Fibonacci n-step number sequences
You are encouraged to solve this task according to the task description, using any language you may know.
These number series are an expansion of the ordinary Fibonacci sequence where:
- For we have the Fibonacci sequence; with initial values and
- For we have the tribonacci sequence; with initial values and
- For we have the tetranacci sequence; with initial values and
... - For general we have the Fibonacci -step sequence - ; with initial values of the first values of the 'th Fibonacci -step sequence ; and 'th value of this 'th sequence being
For small values of , Greek numeric prefixes are sometimes used to individually name each series.
Fibonacci -step sequences Series name Values 2 fibonacci 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ... 3 tribonacci 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ... 4 tetranacci 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ... 5 pentanacci 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ... 6 hexanacci 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ... 7 heptanacci 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ... 8 octonacci 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ... 9 nonanacci 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ... 10 decanacci 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
Allied sequences can be generated where the initial values are changed:
- The Lucas series sums the two preceding values like the fibonacci series for but uses as its initial values.
- Task
- Write a function to generate Fibonacci -step number sequences given its initial values and assuming the number of initial values determines how many previous values are summed to make the next number of the series.
- Use this to print and show here at least the first ten members of the Fibo/tribo/tetra-nacci and Lucas sequences.
- Related tasks
- Also see
- Lucas Numbers - Numberphile (Video)
- Tribonacci Numbers (and the Rauzy Fractal) - Numberphile (Video)
- Wikipedia, Lucas number
- MathWorld, Fibonacci Number
- Some identities for r-Fibonacci numbers
- OEIS Fibonacci numbers
- OEIS Lucas numbers
11l
T Fiblike
Int addnum
[Int] memo
F (start)
.addnum = start.len
.memo = copy(start)
F ()(n)
X.try
R .memo[n]
X.catch IndexError
V ans = sum((n - .addnum .< n).map(i -> (.)(i)))
.memo.append(ans)
R ans
V fibo = Fiblike([1, 1])
print((0.<10).map(i -> fibo(i)))
V lucas = Fiblike([2, 1])
print((0.<10).map(i -> lucas(i)))
L(n, name) zip(2..10, ‘fibo tribo tetra penta hexa hepta octo nona deca’.split(‘ ’))
V fibber = Fiblike([1] [+] (0 .< n - 1).map(i -> Int(2 ^ i)))
print(‘n=#2, #5nacci -> #. ...’.format(n, name, (0.<15).map(i -> String(@fibber(i))).join(‘ ’)))- Output:
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55] [2, 1, 3, 4, 7, 11, 18, 29, 47, 76] n= 2, fibonacci -> 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ... n= 3, tribonacci -> 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ... n= 4, tetranacci -> 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ... n= 5, pentanacci -> 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ... n= 6, hexanacci -> 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ... n= 7, heptanacci -> 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ... n= 8, octonacci -> 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ... n= 9, nonanacci -> 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ... n=10, decanacci -> 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
360 Assembly
* Fibonacci n-step number sequences - 14/04/2020
FIBONS CSECT
USING FIBONS,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
SAVE (14,12) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
LA R6,2 i=2
DO WHILE=(C,R6,LE,=F'7') do i=2 to 7
ST R6,IR ir=i
IF C,R6,EQ,=F'7' THEN if i=7 then - Lucas
LA R0,2 2
ST R0,IR ir=2
ENDIF , endif
LA R0,1 1
ST R0,T t(1)=1
IF C,R6,EQ,=F'7' THEN if i=7 then - Lucas
LA R0,2 2
ST R0,T t(1)=2
ENDIF , endif
LA R0,1 1
ST R0,T+4 t(2)=1
LA R7,3 j=3
DO WHILE=(C,R7,LE,=A(NMAX)) do j=3 to nmax
SR R0,R0 0
ST R0,SUM sum=0
LR R11,R7 j
S R11,IR j-ir
LR R8,R7 k=j
BCTR R8,0 k=j-1
DO WHILE=(CR,R8,GE,R11) do k=j-1 to j-ir by -1
IF LTR,R8,P,R8 THEN if k>0 then
LR R1,R8 k
SLA R1,2 ~
L R2,T-4(R1) t(k)
L R1,SUM sum
AR R1,R2 +
ST R1,SUM sum=sum+t(k)
ENDIF , endif
BCTR R8,0 k--
ENDDO , enddo k
L R0,SUM sum
LR R1,R7 j
SLA R1,2 ~
ST R0,T-4(R1) t(j)=sum
LA R7,1(R7) j++
ENDDO , enddo j
MVC PG,=CL120' ' clear buffer
LA R9,PG @buffer
LR R1,R6 i
BCTR R1,0 i-1
MH R1,=H'5' ~
LA R4,BONACCI-5(R1) @bonacci(i-1)
MVC 0(5,R9),0(R4) output bonacci(i-1)
LA R9,5(R9) @buffer
IF C,R6,NE,=F'7' THEN if i<>7 then
MVC 0(7,R9),=C'nacci: ' output 'nacci: '
ELSE , else
MVC 0(7,R9),=C' : ' output ' : '
ENDIF , endif
LA R9,7(R9) @buffer
LA R7,1 j=1
DO WHILE=(C,R7,LE,=A(NMAX)) do j=1 to nmax
LR R1,R7 j
SLA R1,2 ~
L R2,T-4(R1) t(j)
XDECO R2,XDEC edit t(j)
MVC 0(6,R9),XDEC+6 output t(j)
LA R9,6(R9) @buffer
LA R7,1(R7) j++
ENDDO , enddo j
XPRNT PG,L'PG print buffer
LA R6,1(R6) i++
ENDDO , enddo i
L R13,4(0,R13) restore previous savearea pointer
RETURN (14,12),RC=0 restore registers from calling sav
NMAX EQU 18 sequence length
BONACCI DC CL5' fibo',CL5'tribo',CL5'tetra',CL5'penta',CL5' hexa'
DC CL5'lucas' bonacci(6)
IR DS F ir
SUM DS F sum
T DS (NMAX)F t(nmax)
XDEC DS CL12 temp for xdeco
PG DS CL120 buffer
REGEQU
END FIBONS- Output:
fibonacci: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 tribonacci: 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 tetranacci: 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 pentanacci: 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624 26784 52656 hexanacci: 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 15109 29970 59448 lucas : 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571
ACL2
(defun sum (xs)
(if (endp xs)
0
(+ (first xs)
(sum (rest xs)))))
(defun n-bonacci (prevs limit)
(if (zp limit)
nil
(let ((next (append (rest prevs)
(list (sum prevs)))))
(cons (first next)
(n-bonacci next (1- limit))))))
Output:
> (n-bonacci '(1 1) 10) (1 2 3 5 8 13 21 34 55 89) > (n-bonacci '(1 1 2) 10) (1 2 4 7 13 24 44 81 149 274) > (n-bonacci '(1 1 2 4) 10) (1 2 4 8 15 29 56 108 208 401) > (n-bonacci '(2 1) 10) (1 3 4 7 11 18 29 47 76 123)
Action!
DEFINE MAX="15"
PROC GenerateSeq(CARD ARRAY init BYTE nInit CARD ARRAY seq BYTE nSeq)
CARD next
BYTE i,j,n
IF nInit<nSeq THEN
n=nInit
ELSE
n=nSeq
FI
FOR i=0 TO n-1
DO
seq(i)=init(i)
OD
FOR i=n TO nSeq-1
DO
next=0
FOR j=i-nInit TO i-1
DO
next==+seq(j)
OD
seq(i)=next
OD
RETURN
PROC PrintSeq(CHAR ARRAY name CARD ARRAY seq BYTE n)
BYTE i
PrintF("%S=[",name)
FOR i=0 TO n-1
DO
PrintC(seq(i))
IF i<n-1 THEN
Print(" ")
ELSE
PrintE("]")
FI
OD
RETURN
PROC SetInverseVideo(CHAR ARRAY text)
BYTE i
FOR i=1 TO text(0)
DO
text(i)=text(i) OR $80
OD
RETURN
PROC Test(CHAR ARRAY name CARD ARRAY init CARD ARRAY nInit BYTE nSeq)
CARD ARRAY seq(MAX)
SetInverseVideo(name)
GenerateSeq(init,nInit,seq,nSeq)
PrintSeq(name,seq,nSeq)
RETURN
PROC Main()
CARD ARRAY fibInit=[1 1 2 4 8 16 32 64 128 256 512]
CARD ARRAY lucInit=[2 1]
Test("lucas",lucInit,2,MAX)
Test("fibonacci",fibInit,2,MAX)
Test("tribonacci",fibInit,3,MAX)
Test("tetranacci",fibInit,4,MAX)
Test("pentanacci",fibInit,5,MAX)
Test("hexanacci",fibInit,6,MAX)
Test("heptanacci",fibInit,7,MAX)
Test("octanacci",fibInit,8,MAX)
Test("nonanacci",fibInit,9,MAX)
Test("decanacci",fibInit,10,MAX)
RETURN- Output:
Screenshot from Atari 8-bit computer
lucas=[2 1 3 4 7 11 18 29 47 76 123 19 9 322 521 843] fibonacci=[1 1 2 3 5 8 13 21 34 55 89 144 233 377 610] tribonacci=[1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136] tetranacci=[1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536] pentanacci=[1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930] hexanacci=[1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617] heptanacci=[1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936] octanacci=[1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080] nonanacci=[1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144] decanacci=[1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172]
Ada
First, we specify a package Bonacci, that defines the type Sequence (of Positive numbers), a function Generate that takes a given Start sequence and outputs a generalized N-Bonacci Sequence of a spefified Length, and some constant start sequences.
package Bonacci is
type Sequence is array(Positive range <>) of Positive;
function Generate(Start: Sequence; Length: Positive := 10) return Sequence;
Start_Fibonacci: constant Sequence := (1, 1);
Start_Tribonacci: constant Sequence := (1, 1, 2);
Start_Tetranacci: constant Sequence := (1, 1, 2, 4);
Start_Lucas: constant Sequence := (2, 1);
end Bonacci;
The implementation is quite straightforward.
package body Bonacci is
function Generate(Start: Sequence; Length: Positive := 10) return Sequence is
begin
if Length <= Start'Length then
return Start(Start'First .. Start'First+Length-1);
else
declare
Sum: Natural := 0;
begin
for I in Start'Range loop
Sum := Sum + Start(I);
end loop;
return Start(Start'First)
& Generate(Start(Start'First+1 .. Start'Last) & Sum, Length-1);
end;
end if;
end Generate;
end Bonacci;
Finally, we actually generate some sequences, as required by the task. For convenience, we define a procedure Print that outputs a sequence,
with Ada.Text_IO, Bonacci;
procedure Test_Bonacci is
procedure Print(Name: String; S: Bonacci.Sequence) is
begin
Ada.Text_IO.Put(Name & "(");
for I in S'First .. S'Last-1 loop
Ada.Text_IO.Put(Integer'Image(S(I)) & ",");
end loop;
Ada.Text_IO.Put_Line(Integer'Image(S(S'Last)) & " )");
end Print;
begin
Print("Fibonacci: ", Bonacci.Generate(Bonacci.Start_Fibonacci));
Print("Tribonacci: ", Bonacci.Generate(Bonacci.Start_Tribonacci));
Print("Tetranacci: ", Bonacci.Generate(Bonacci.Start_Tetranacci));
Print("Lucas: ", Bonacci.Generate(Bonacci.Start_Lucas));
Print("Decanacci: ",
Bonacci.Generate((1, 1, 2, 4, 8, 16, 32, 64, 128, 256), 15));
end Test_Bonacci;
The output:
Fibonacci: ( 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ) Tribonacci: ( 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 ) Tetranacci: ( 1, 1, 2, 4, 8, 15, 29, 56, 108, 208 ) Lucas: ( 2, 1, 3, 4, 7, 11, 18, 29, 47, 76 ) Decanacci: ( 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172 )
ALGOL 68
# returns an array of the first required count elements of an a n-step fibonacci sequence #
# the initial values are taken from the init array #
PROC n step fibonacci sequence = ( []INT init, INT required count )[]INT:
BEGIN
[ 1 : required count ]INT result;
[]INT initial values = init[ AT 1 ];
INT step = UPB initial values;
# install the initial values #
FOR n TO step DO result[ n ] := initial values[ n ] OD;
# calculate the rest of the sequence #
FOR n FROM step + 1 TO required count DO
result[ n ] := 0;
FOR p FROM n - step TO n - 1 DO result[ n ] +:= result[ p ] OD
OD;
result
END; # required count #
# prints the elements of a sequence #
PROC print sequence = ( STRING legend, []INT sequence )VOID:
BEGIN
print( ( legend, ":" ) );
FOR e FROM LWB sequence TO UPB sequence DO print( ( " ", whole( sequence[ e ], 0 ) ) ) OD;
print( ( newline ) )
END; # print sequence #
# print some sequences #
print sequence( "fibonacci ", n step fibonacci sequence( ( 1, 1 ), 10 ) );
print sequence( "tribonacci ", n step fibonacci sequence( ( 1, 1, 2 ), 10 ) );
print sequence( "tetrabonacci", n step fibonacci sequence( ( 1, 1, 2, 4 ), 10 ) );
print sequence( "lucus ", n step fibonacci sequence( ( 2, 1 ), 10 ) )- Output:
fibonacci : 1 1 2 3 5 8 13 21 34 55 tribonacci : 1 1 2 4 7 13 24 44 81 149 tetrabonacci: 1 1 2 4 8 15 29 56 108 208 lucus : 2 1 3 4 7 11 18 29 47 76
APL
nStep ← {⊃(1↓⊢,+/)⍣(⍺-1)⊢⍵}
nacci ← 2*0⌈¯2+⍳
↑((⍳10)nStep¨⊂)¨(nacci¨2 3 4),⊂2 1
- Output:
1 1 2 3 5 8 13 21 34 55 1 1 2 4 7 13 24 44 81 149 1 1 2 4 8 15 29 56 108 208 2 1 3 4 7 11 18 29 47 76
AppleScript
Functional
use AppleScript version "2.4"
use framework "Foundation"
use scripting additions
-- Start sequence -> Number of terms -> terms
-- takeNFibs :: [Int] -> Int -> [Int]
on takeNFibs(xs, n)
script go
on |λ|(xs, n)
if 0 < n and 0 < length of xs then
cons(head(xs), ¬
|λ|(append(tail(xs), {sum(xs)}), n - 1))
else
{}
end if
end |λ|
end script
go's |λ|(xs, n)
end takeNFibs
-- fibInit :: Int -> [Int]
on fibInit(n)
script powerOfTwo
on |λ|(x)
2 ^ x as integer
end |λ|
end script
cons(1, map(powerOfTwo, enumFromToInt(0, n - 2)))
end fibInit
-- TEST ---------------------------------------------------
on run
set intTerms to 15
script series
on |λ|(s, n)
justifyLeft(12, space, s & "nacci") & " -> " & ¬
showJSON(takeNFibs(fibInit(n), intTerms))
end |λ|
end script
set strTable to unlines(zipWith(series, ¬
words of ("fibo tribo tetra penta hexa hepta octo nona deca"), ¬
enumFromToInt(2, 10)))
justifyLeft(12, space, "Lucas ") & " -> " & ¬
showJSON(takeNFibs({2, 1}, intTerms)) & linefeed & strTable
end run
-- GENERIC FUNCTIONS --------------------------------------
-- Append two lists.
-- append (++) :: [a] -> [a] -> [a]
-- append (++) :: String -> String -> String
on append(xs, ys)
xs & ys
end append
-- cons :: a -> [a] -> [a]
on cons(x, xs)
if list is class of xs then
{x} & xs
else
x & xs
end if
end cons
-- enumFromToInt :: Int -> Int -> [Int]
on enumFromToInt(m, n)
if m ≤ n then
set lst to {}
repeat with i from m to n
set end of lst to i
end repeat
return lst
else
return {}
end if
end enumFromToInt
-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- head :: [a] -> a
on head(xs)
if xs = {} then
missing value
else
item 1 of xs
end if
end head
-- justifyLeft :: Int -> Char -> String -> String
on justifyLeft(n, cFiller, strText)
if n > length of strText then
text 1 thru n of (strText & replicate(n, cFiller))
else
strText
end if
end justifyLeft
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- min :: Ord a => a -> a -> a
on min(x, y)
if y < x then
y
else
x
end if
end min
-- Egyptian multiplication - progressively doubling a list, appending
-- stages of doubling to an accumulator where needed for binary
-- assembly of a target length
-- replicate :: Int -> a -> [a]
on replicate(n, a)
set out to {}
if n < 1 then return out
set dbl to {a}
repeat while (n > 1)
if (n mod 2) > 0 then set out to out & dbl
set n to (n div 2)
set dbl to (dbl & dbl)
end repeat
return out & dbl
end replicate
-- showJSON :: a -> String
on showJSON(x)
set c to class of x
if (c is list) or (c is record) then
set ca to current application
set {json, e} to ca's NSJSONSerialization's ¬
dataWithJSONObject:x options:0 |error|:(reference)
if json is missing value then
e's localizedDescription() as text
else
(ca's NSString's alloc()'s ¬
initWithData:json encoding:(ca's NSUTF8StringEncoding)) as text
end if
else if c is date then
"\"" & ((x - (time to GMT)) as «class isot» as string) & ".000Z" & "\""
else if c is text then
"\"" & x & "\""
else if (c is integer or c is real) then
x as text
else if c is class then
"null"
else
try
x as text
on error
("«" & c as text) & "»"
end try
end if
end showJSON
-- sum :: [Num] -> Num
on sum(xs)
script add
on |λ|(a, b)
a + b
end |λ|
end script
foldl(add, 0, xs)
end sum
-- tail :: [a] -> [a]
on tail(xs)
set blnText to text is class of xs
if blnText then
set unit to ""
else
set unit to {}
end if
set lng to length of xs
if 1 > lng then
missing value
else if 2 > lng then
unit
else
if blnText then
text 2 thru -1 of xs
else
rest of xs
end if
end if
end tail
-- unlines :: [String] -> String
on unlines(xs)
set {dlm, my text item delimiters} to ¬
{my text item delimiters, linefeed}
set str to xs as text
set my text item delimiters to dlm
str
end unlines
-- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
on zipWith(f, xs, ys)
set lng to min(length of xs, length of ys)
if 1 > lng then return {}
set lst to {}
tell mReturn(f)
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, item i of ys)
end repeat
return lst
end tell
end zipWith
- Output:
Lucas -> [2,1,3,4,7,11,18,29,47,76,123,199,322,521,843] fibonacci -> [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610] tribonacci -> [1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136] tetranacci -> [1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536] pentanacci -> [1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930] hexanacci -> [1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617] heptanacci -> [1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936] octonacci -> [1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080] nonanacci -> [1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144] decanacci -> [1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172]
Simple
-- Parameters:
-- n: …nacci step size as integer. Alternatively "Lucas".
-- F: Maximum …nacci index required. (0-based.)
on fibonacciNStep(n, F)
script o
property sequence : {0}
end script
if (n is "Lucas") then set {n, item 1 of o's sequence} to {2, 2}
-- F1 (if included) is always 1.
if (F > 0) then set end of o's sequence to 1
-- F2 (ditto) is F0 + F1.
if (F > 1) then set end of o's sequence to (beginning of o's sequence) + (end of o's sequence)
-- Each further number up to and including Fn is twice the number preceding it.
if (n > F) then set n to F
repeat (n - 2) times
set end of o's sequence to (end of o's sequence) * 2
end repeat
-- Beyond Fn, each number is twice the one preceding it, minus the number n places before that.
set nBeforeEnd to -(n + 1)
repeat (F - n) times
set end of o's sequence to (end of o's sequence) * 2 - (item nBeforeEnd of o's sequence)
end repeat
return o's sequence
end fibonacciNStep
-- Test code:
set maxF to 15 -- Length of sequence required after the initial 0 or 2.
set seriesNames to {missing value, "fibonacci: ", "tribonacci: ", "tetranacci: ", "pentanacci: ", ¬
"hexanacci: ", "heptanacci: ", "octonacci: ", "nonanacci: ", "decanacci: "}
set output to {}
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to ", "
repeat with nacciSize from 2 to 10
set end of output to (item nacciSize of seriesNames) & fibonacciNStep(nacciSize, maxF) & " …"
end repeat
set end of output to "Lucas: " & fibonacciNStep("lucas", maxF) & " …"
set AppleScript's text item delimiters to linefeed
set output to output as text
set AppleScript's text item delimiters to astid
return output
- Output:
"fibonacci: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 …
tribonacci: 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136 …
tetranacci: 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536 …
pentanacci: 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930 …
hexanacci: 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617 …
heptanacci: 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936 …
octonacci: 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080 …
nonanacci: 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144 …
decanacci: 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172 …
Lucas: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364 …"
Arturo
naccis: #[
lucas: [2 1]
fibonacci: [1 1]
tribonacci: [1 1 2]
tetranacci: [1 1 2 4]
pentanacci: [1 1 2 4 8]
hexanacci: [1 1 2 4 8 16]
heptanacci: [1 1 2 4 8 16 32]
octonacci: [1 1 2 4 8 16 32 64]
nonanacci: [1 1 2 4 8 16 32 64 128]
decanacci: [1 1 2 4 8 16 32 64 128 256]
]
anyNacci: function [start, count][
n: size start
result: new start
do.times: count-n ->
result: result ++ sum last.n:n result
return join.with:", " to [:string] result
]
loop naccis [k,v][
print [pad (k ++ ":") 12 anyNacci v 15]
]
- Output:
lucas: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 tribonacci: 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136 tetranacci: 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536 pentanacci: 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930 hexanacci: 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617 heptanacci: 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936 octonacci: 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080 nonanacci: 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144 decanacci: 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172
AutoHotkey
for i, seq in ["nacci", "lucas"]
Loop, 9 {
Out .= seq "(" A_Index + 1 "): "
for key, val in NStepSequence(i, 1, A_Index + 1, 15)
Out .= val (A_Index = 15 ? "`n" : "`, ")
}
MsgBox, % Out
NStepSequence(v1, v2, n, k) {
a := [v1, v2]
Loop, % k - 2 {
a[j := A_Index + 2] := 0
Loop, % j < n + 2 ? j - 1 : n
a[j] += a[j - A_Index]
}
return, a
}
Output:
nacci(2): 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 nacci(3): 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136 nacci(4): 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536 nacci(5): 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930 nacci(6): 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617 nacci(7): 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936 nacci(8): 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080 nacci(9): 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144 nacci(10): 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172 lucas(2): 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 lucas(3): 2, 1, 3, 6, 10, 19, 35, 64, 118, 217, 399, 734, 1350, 2483, 4567 lucas(4): 2, 1, 3, 6, 12, 22, 43, 83, 160, 308, 594, 1145, 2207, 4254, 8200 lucas(5): 2, 1, 3, 6, 12, 24, 46, 91, 179, 352, 692, 1360, 2674, 5257, 10335 lucas(6): 2, 1, 3, 6, 12, 24, 48, 94, 187, 371, 736, 1460, 2896, 5744, 11394 lucas(7): 2, 1, 3, 6, 12, 24, 48, 96, 190, 379, 755, 1504, 2996, 5968, 11888 lucas(8): 2, 1, 3, 6, 12, 24, 48, 96, 192, 382, 763, 1523, 3040, 6068, 12112 lucas(9): 2, 1, 3, 6, 12, 24, 48, 96, 192, 384, 766, 1531, 3059, 6112, 12212 lucas(10): 2, 1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1534, 3067, 6131, 12256
AWK
function sequence(values, howmany) {
init_length = length(values)
for (i=init_length + 1; i<=howmany; i++) {
values[i] = 0
for (j=1; j<=init_length; j++) {
values[i] += values[i-j]
}
}
result = ""
for (i in values) {
result = result values[i] " "
}
delete values
return result
}
# print some sequences
END {
a[1] = 1; a[2] = 1
print("fibonacci :\t",sequence(a, 10))
a[1] = 1; a[2] = 1; a[3] = 2
print("tribonacci :\t",sequence(a, 10))
a[1] = 1 ; a[2] = 1 ; a[3] = 2 ; a[4] = 4
print("tetrabonacci :\t",sequence(a, 10))
a[1] = 2; a[2] = 1
print("lucas :\t\t",sequence(a, 10))
}
Output:
fibonacci : 1 1 2 3 5 8 13 21 34 55 tribonacci : 1 1 2 4 7 13 24 44 81 149 tetrabonacci : 1 1 2 4 8 15 29 56 108 208 lucas : 2 1 3 4 7 11 18 29 47 76
BASIC
BASIC256
# Rosetta Code problem: https://www.rosettacode.org/wiki/Fibonacci_n-step_number_sequences
# by Jjuanhdez, 06/2022
arraybase 1
print " fibonacci =>";
dim a = {1,1}
call fib (a)
print " tribonacci =>";
dim a = {1,1,2}
call fib (a)
print " tetranacci =>";
dim a = {1,1,2,4}
call fib (a)
print " pentanacci =>";
dim a = {1,1,2,4,8}
call fib (a)
print " hexanacci =>";
dim a = {1,1,2,4,8,16}
call fib (a)
print " heptanacci =>";
dim a = {1,1,2,4,8,16,32}
call fib (a)
print " octonacci =>";
dim a = {1,1,2,4,8,16,32,64}
call fib (a)
print " nonanacci =>";
dim a = {1,1,2,4,8,16,32,64,128}
call fib (a)
print " decanacci =>";
dim a = {1,1,2,4,8,16,32,64,128,256}
call fib (a)
print " lucas =>";
dim a = {2,1}
call fib (a)
end
subroutine fib (a)
dim f(24) fill 0
b = 0
for x = 1 to a[?]
b += 1
f[x] = a[x]
next x
for i = b to 13 + b
print rjust(f[i-b+1], 5);
if i <> 13 + b then print ","; else print ", ..."
for j = (i-b+1) to i
f[i+1] = f[i+1] + f[j]
next j
next i
end subroutine- Output:
fibonacci => 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...
tribonacci => 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, ...
tetranacci => 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, ...
pentanacci => 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, ...
hexanacci => 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, ...
heptanacci => 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, ...
octonacci => 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, ...
nonanacci => 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, ...
decanacci => 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, ...
lucas => 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, ...
Chipmunk Basic
100 sub fib(a())
110 erase f
120 dim f(24)
130 b = 0
140 for x = 1 to ubound(a)
150 b = b+1
160 f(x) = a(x)
170 next x
180 for i = b to 12+b
190 print using "#### ";f(i-b+1);
200 for j = (i-b+1) to i
210 f(i+1) = f(i+1)+f(j)
220 next j
230 next i
240 print
250 end sub
260 cls
270 print " fibonacci =>";
280 dim a(2)
290 a(1) = 1 : a(2) = 1
300 fib(a())
310 print " tribonacci =>";
320 dim a(3)
330 a(1) = 1 : a(2) = 1 : a(3) = 2
340 fib(a())
350 print " tetranacci =>";
360 dim c(4)
370 c(1) = 1 : c(2) = 1 : c(3) = 2 : c(4) = 4
380 fib(c())
390 print " lucas =>";
400 dim d(2)
410 d(1) = 2 : d(2) = 1
420 fib(d())
430 end
QBasic
DECLARE SUB fib (a() AS INTEGER)
CLS
PRINT " fibonacci =>";
DIM a(1 TO 2) AS INTEGER
a(1) = 1: a(2) = 1
CALL fib(a())
PRINT " tribonacci =>";
DIM b(1 TO 3) AS INTEGER
b(1) = 1: b(2) = 1: b(3) = 2
CALL fib(b())
PRINT " tetranacci =>";
DIM c(1 TO 4) AS INTEGER
c(1) = 1: c(2) = 1: c(3) = 2: c(4) = 4
CALL fib(c())
PRINT " lucas =>";
DIM d(1 TO 2) AS INTEGER
d(1) = 2: d(2) = 1
CALL fib(d())
END
SUB fib (a() AS INTEGER)
DIM f(24)
b = 0
FOR x = 1 TO UBOUND(a)
b = b + 1
f(x) = a(x)
NEXT x
FOR i = b TO 12 + b
PRINT USING "#### "; f(i - b + 1);
FOR j = (i - b + 1) TO i
f(i + 1) = f(i + 1) + f(j)
NEXT j
NEXT i
END SUB
QB64
Rem $Dynamic
Cls
Print " fibonacci =>";
Dim a(1 To 2) As Integer
a(1) = 1
a(2) = 1
Call fib(a())
Print " tribonacci =>";
ReDim _Preserve a(1 To 3)
a(3) = 2
Call fib(a())
Print " tetranacci =>";
ReDim _Preserve a(1 To 4)
a(4) = 4
Call fib(a())
Print " lucas =>";
ReDim a(1 To 2)
a(1) = 2
a(2) = 1
Call fib(a())
End
Sub fib (a() As Integer)
Dim f(24)
b = 0
For x = 1 To UBound(a)
b = b + 1
f(x) = a(x)
Next x
For i = b To 12 + b
Print Using "#### "; f(i - b + 1);
For j = (i - b + 1) To i
f(i + 1) = f(i + 1) + f(j)
Next j
Next i
End Sub
Batch File
@echo off
echo Fibonacci Sequence:
call:nfib 1 1
echo.
echo Tribonacci Sequence:
call:nfib 1 1 2
echo.
echo Tetranacci Sequence:
call:nfib 1 1 2 4
echo.
echo Lucas Numbers:
call:nfib 2 1
echo.
pause>nul
exit /b
:nfib
setlocal enabledelayedexpansion
for %%i in (%*) do (
set /a count+=1
set seq=!seq! %%i
)
set "seq=%seq% ^| "
set n=-%count%
set /a n+=1
for %%i in (%*) do (
set F!n!=%%i
set /a n+=1
)
for /l %%i in (1,1,10) do (
set /a termstart=%%i-%count%%
set /a termend=%%i-1
for /l %%j in (!termstart!,1,!termend!) do (
set /a F%%i+=!F%%j!
)
set seq=!seq! !F%%i!
)
echo %seq%
endlocal
exit /b
- Output:
Fibonacci Sequence: 1 1 | 2 3 5 8 13 21 34 55 89 144 Tribonacci Sequence: 1 1 2 | 4 7 13 24 44 81 149 274 504 927 Tetranacci Sequence: 1 1 2 4 | 8 15 29 56 108 208 401 773 1490 2872 Lucas Numbers: 2 1 | 3 4 7 11 18 29 47 76 123 199
BBC BASIC
The BBC BASIC SUM function is useful here.
@% = 5 : REM Column width
PRINT "Fibonacci:"
DIM f2%(1) : f2%() = 1,1
FOR i% = 1 TO 12 : PRINT f2%(0); : PROCfibn(f2%()) : NEXT : PRINT " ..."
PRINT "Tribonacci:"
DIM f3%(2) : f3%() = 1,1,2
FOR i% = 1 TO 12 : PRINT f3%(0); : PROCfibn(f3%()) : NEXT : PRINT " ..."
PRINT "Tetranacci:"
DIM f4%(3) : f4%() = 1,1,2,4
FOR i% = 1 TO 12 : PRINT f4%(0); : PROCfibn(f4%()) : NEXT : PRINT " ..."
PRINT "Lucas:"
DIM fl%(1) : fl%() = 2,1
FOR i% = 1 TO 12 : PRINT fl%(0); : PROCfibn(fl%()) : NEXT : PRINT " ..."
END
DEF PROCfibn(f%())
LOCAL i%, s%
s% = SUM(f%())
FOR i% = 1 TO DIM(f%(),1)
f%(i%-1) = f%(i%)
NEXT
f%(i%-1) = s%
ENDPROC
Output:
Fibonacci:
1 1 2 3 5 8 13 21 34 55 89 144 ...
Tribonacci:
1 1 2 4 7 13 24 44 81 149 274 504 ...
Tetranacci:
1 1 2 4 8 15 29 56 108 208 401 773 ...
Lucas:
2 1 3 4 7 11 18 29 47 76 123 199 ...
Befunge
110p>>55+109"iccanaceD"22099v
v9013"Tetranacci"9014"Lucas"<
>"iccanobirT"2109"iccanobiF"v
>>:#,_0p20p0>:01-\2>#v0>#g<>>
^_@#:,+55$_^ JH v`1:v#\p03<
_$.1+:77+`^vg03:_0g+>\:1+#^
50p-\30v v\<>\30g1-\^$$_:1-
05g04\g< >`#^_:40p30g0>^!:g
- Output:
Fibonacci 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 Tribonacci 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 Tetranacci 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 Lucas 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 Decanacci 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172
Bracmat
( ( nacci
= Init Cnt N made tail
. ( plus
= n
. !arg:#%?n ?arg&!n+plus$!arg
| 0
)
& !arg:(?Init.?Cnt)
& !Init:? [?N
& !Init:?made
& !Cnt+-1*!N:?times
& -1+-1*!N:?M
& whl
' ( !times+-1:~<0:?times
& !made:? [!M ?tail
& !made plus$!tail:?made
)
& !made
)
& ( pad
= len w
. @(!arg:? [?len)
& @(" ":? [!len ?w)
& !w !arg
)
& (fibonacci.1 1)
(tribonacci.1 1 2)
(tetranacci.1 1 2 4)
(pentanacci.1 1 2 4 8)
(hexanacci.1 1 2 4 8 16)
(heptanacci.1 1 2 4 8 16 32)
(octonacci.1 1 2 4 8 16 32 64)
(nonanacci.1 1 2 4 8 16 32 64 128)
(decanacci.1 1 2 4 8 16 32 64 128 256)
(lucas.2 1)
: ?L
& whl
' ( !L:(?name.?Init) ?L
& out$(str$(pad$!name ": ") nacci$(!Init.12))
)
);Output:
fibonacci: 1 1 2 3 5 8 13 21 34 55 89 144
tribonacci: 1 1 2 4 7 13 24 44 81 149 274 504
tetranacci: 1 1 2 4 8 15 29 56 108 208 401 773
pentanacci: 1 1 2 4 8 16 31 61 120 236 464 912
hexanacci: 1 1 2 4 8 16 32 63 125 248 492 976
heptanacci: 1 1 2 4 8 16 32 64 127 253 504 1004
octonacci: 1 1 2 4 8 16 32 64 128 255 509 1016
nonanacci: 1 1 2 4 8 16 32 64 128 256 511 1021
decanacci: 1 1 2 4 8 16 32 64 128 256 512 1023
lucas: 2 1 3 4 7 11 18 29 47 76 123 199
BQN
NStep ← ⊑(1↓⊢∾+´)∘⊢⍟⊣
Nacci ← (2⋆0∾↕)∘(⊢-1˙)
>((↕10) NStep¨ <)¨ (Nacci¨ 2‿3‿4) ∾ <2‿1
- Output:
┌─
╵ 1 1 2 3 5 8 13 21 34 55
1 1 2 4 7 13 24 44 81 149
1 1 2 4 8 15 29 56 108 208
2 1 3 4 7 11 18 29 47 76
┘
C
/*
The function anynacci determines the n-arity of the sequence from the number of seed elements. 0 ended arrays are used since C does not have a way of determining the length of dynamic and function-passed integer arrays.*/
#include<stdlib.h>
#include<stdio.h>
int *
anynacci (int *seedArray, int howMany)
{
int *result = malloc (howMany * sizeof (int));
int i, j, initialCardinality;
for (i = 0; seedArray[i] != 0; i++);
initialCardinality = i;
for (i = 0; i < initialCardinality; i++)
result[i] = seedArray[i];
for (i = initialCardinality; i < howMany; i++)
{
result[i] = 0;
for (j = i - initialCardinality; j < i; j++)
result[i] += result[j];
}
return result;
}
int
main ()
{
int fibo[] = { 1, 1, 0 }, tribo[] = { 1, 1, 2, 0 }, tetra[] = { 1, 1, 2, 4, 0 }, luca[] = { 2, 1, 0 };
int *fibonacci = anynacci (fibo, 10), *tribonacci = anynacci (tribo, 10), *tetranacci = anynacci (tetra, 10),
*lucas = anynacci(luca, 10);
int i;
printf ("\nFibonacci\tTribonacci\tTetranacci\tLucas\n");
for (i = 0; i < 10; i++)
printf ("\n%d\t\t%d\t\t%d\t\t%d", fibonacci[i], tribonacci[i],
tetranacci[i], lucas[i]);
return 0;
}
Output:
Fibonacci Tribonacci Tetranacci Lucas 1 1 1 2 1 1 1 1 2 2 2 3 3 4 4 4 5 7 8 7 8 13 15 11 13 24 29 18 21 44 56 29 34 81 108 47 55 149 208 76
C#
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace Fibonacci
{
class Program
{
static void Main(string[] args)
{
PrintNumberSequence("Fibonacci", GetNnacciNumbers(2, 10));
PrintNumberSequence("Lucas", GetLucasNumbers(10));
PrintNumberSequence("Tribonacci", GetNnacciNumbers(3, 10));
PrintNumberSequence("Tetranacci", GetNnacciNumbers(4, 10));
Console.ReadKey();
}
private static IList<ulong> GetLucasNumbers(int length)
{
IList<ulong> seedSequence = new List<ulong>() { 2, 1 };
return GetFibLikeSequence(seedSequence, length);
}
private static IList<ulong> GetNnacciNumbers(int seedLength, int length)
{
return GetFibLikeSequence(GetNacciSeed(seedLength), length);
}
private static IList<ulong> GetNacciSeed(int seedLength)
{
IList<ulong> seedSquence = new List<ulong>() { 1 };
for (uint i = 0; i < seedLength - 1; i++)
{
seedSquence.Add((ulong)Math.Pow(2, i));
}
return seedSquence;
}
private static IList<ulong> GetFibLikeSequence(IList<ulong> seedSequence, int length)
{
IList<ulong> sequence = new List<ulong>();
int count = seedSequence.Count();
if (length <= count)
{
sequence = seedSequence.Take((int)length).ToList();
}
else
{
sequence = seedSequence;
for (int i = count; i < length; i++)
{
ulong num = 0;
for (int j = 0; j < count; j++)
{
num += sequence[sequence.Count - 1 - j];
}
sequence.Add(num);
}
}
return sequence;
}
private static void PrintNumberSequence(string Title, IList<ulong> numbersequence)
{
StringBuilder output = new StringBuilder(Title).Append(" ");
foreach (long item in numbersequence)
{
output.AppendFormat("{0}, ", item);
}
Console.WriteLine(output.ToString());
}
}
}
Fibonacci 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, Lucas 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, Tribonacci 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, Tetranacci 1, 1, 2, 4, 8, 15, 29, 56, 108, 208,
C++
#include <vector>
#include <iostream>
#include <numeric>
#include <iterator>
#include <memory>
#include <string>
#include <algorithm>
#include <iomanip>
std::vector<int> nacci ( const std::vector<int> & start , int arity ) {
std::vector<int> result ( start ) ;
int sumstart = 1 ;//summing starts at vector's begin + sumstart as
//soon as the vector is longer than arity
while ( result.size( ) < 15 ) { //we print out the first 15 numbers
if ( result.size( ) <= arity )
result.push_back( std::accumulate( result.begin( ) ,
result.begin( ) + result.size( ) , 0 ) ) ;
else {
result.push_back( std::accumulate ( result.begin( ) +
sumstart , result.begin( ) + sumstart + arity , 0 )) ;
sumstart++ ;
}
}
return std::move ( result ) ;
}
int main( ) {
std::vector<std::string> naccinames {"fibo" , "tribo" ,
"tetra" , "penta" , "hexa" , "hepta" , "octo" , "nona" , "deca" } ;
const std::vector<int> fibo { 1 , 1 } , lucas { 2 , 1 } ;
for ( int i = 2 ; i < 11 ; i++ ) {
std::vector<int> numberrow = nacci ( fibo , i ) ;
std::cout << std::left << std::setw( 10 ) <<
naccinames[ i - 2 ].append( "nacci" ) <<
std::setw( 2 ) << " : " ;
std::copy ( numberrow.begin( ) , numberrow.end( ) ,
std::ostream_iterator<int>( std::cout , " " ) ) ;
std::cout << "...\n" ;
numberrow = nacci ( lucas , i ) ;
std::cout << "Lucas-" << i ;
if ( i < 10 ) //for formatting purposes
std::cout << " : " ;
else
std::cout << " : " ;
std::copy ( numberrow.begin( ) , numberrow.end( ) ,
std::ostream_iterator<int>( std::cout , " " ) ) ;
std::cout << "...\n" ;
}
return 0 ;
}
Output:
fibonacci : 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ... Lucas-2 : 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 ... tribonacci : 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ... Lucas-3 : 2 1 3 6 10 19 35 64 118 217 399 734 1350 2483 4567 ... tetranacci : 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ... Lucas-4 : 2 1 3 6 12 22 43 83 160 308 594 1145 2207 4254 8200 ... pentanacci : 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ... Lucas-5 : 2 1 3 6 12 24 46 91 179 352 692 1360 2674 5257 10335 ... hexanacci : 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ... Lucas-6 : 2 1 3 6 12 24 48 94 187 371 736 1460 2896 5744 11394 ... heptanacci : 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ... Lucas-7 : 2 1 3 6 12 24 48 96 190 379 755 1504 2996 5968 11888 ... octonacci : 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ... Lucas-8 : 2 1 3 6 12 24 48 96 192 382 763 1523 3040 6068 12112 ... nonanacci : 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ... Lucas-9 : 2 1 3 6 12 24 48 96 192 384 766 1531 3059 6112 12212 ... decanacci : 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ... Lucas-10 : 2 1 3 6 12 24 48 96 192 384 768 1534 3067 6131 12256 ...
Alternate Version
This version focuses on a clean, simple class that adapts to any pair of starting numbers and any order. Rather than summing over all history every time, it uses an O(1) incremental update to a running total. Thus, performance remains essentially unchanged even for very large orders.
#include <iostream>
#include <vector>
// This class forms a simple 'generator', where operator() returns the next
// element in the series. It uses a small sliding window buffer to minimize
// storage overhead.
class nacci_t
{
std::vector< int > history;
unsigned windex; // sliding window index
unsigned rindex; // result index
int running_sum; // sum of values in sliding window
public:
nacci_t( unsigned int order, int a0 = 1, int a1 = 1 )
: history( order + 1 ), windex( 0 ), rindex( order - 1 ),
running_sum( a0 + a1 )
{
// intialize sliding window
history[order - 1] = a0;
history[order - 0] = a1;
}
int operator()()
{
int result = history[ rindex ]; // get 'nacci number to return
running_sum -= history[ windex ]; // old 'nacci falls out of window
history[ windex ] = running_sum; // new 'nacci enters the window
running_sum += running_sum; // new 'nacci added to the sum
if ( ++windex == history.size() ) windex = 0;
if ( ++rindex == history.size() ) rindex = 0;
return result;
}
};
int main()
{
for ( unsigned int i = 2; i <= 10; ++i )
{
nacci_t nacci( i ); // fibonacci sequence
std::cout << "nacci( " << i << " ): ";
for ( int j = 0; j < 10; ++j )
std::cout << " " << nacci();
std::cout << std::endl;
}
for ( unsigned int i = 2; i <= 10; ++i )
{
nacci_t lucas( i, 2, 1 ); // Lucas sequence
std::cout << "lucas( " << i << " ): ";
for ( int j = 0; j < 10; ++j )
std::cout << " " << lucas();
std::cout << std::endl;
}
}
Clojure
(defn nacci [init]
(letfn [(s [] (lazy-cat init (apply map + (map #(drop % (s)) (range (count init))))))]
(s)))
(let [show (fn [name init] (println "first 20" name (take 20 (nacci init))))]
(show "Fibonacci" [1 1])
(show "Tribonacci" [1 1 2])
(show "Tetranacci" [1 1 2 4])
(show "Lucas" [2 1]))
- Output:
first 20 Fibonacci (1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765) first 20 Tribonacci (1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 35890 66012) first 20 Tetranacci (1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 76424 147312) first 20 Lucas (2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571 5778 9349)
CLU
% Find the Nth element of a given n-step sequence
n_step = proc (seq: sequence[int], n: int) returns (int)
a: array[int] := sequence[int]$s2a(seq)
for i: int in int$from_to(1,n) do
sum: int := 0
for x: int in array[int]$elements(a) do
sum := sum + x
end
array[int]$reml(a)
array[int]$addh(a,sum)
end
return(array[int]$bottom(a))
end n_step
% Generate the initial sequence for the Fibonacci n-step sequence of length N
anynacci = proc (n: int) returns (sequence[int])
a: array[int] := array[int]$[1]
for i: int in int$from_to(0,n-2) do
array[int]$addh(a, 2**i)
end
return(sequence[int]$a2s(a))
end anynacci
% Given an initial sequence, print the first N elements
print_n = proc (seq: sequence[int], n: int)
po: stream := stream$primary_output()
for i: int in int$from_to(0, n-1) do
stream$putright(po, int$unparse(n_step(seq, i)), 4)
end
stream$putl(po, "")
end print_n
start_up = proc ()
s = struct[name: string, seq: sequence[int]]
po: stream := stream$primary_output()
seqs: array[s] := array[s]$[
s${name: "Fibonacci", seq: anynacci(2)},
s${name: "Tribonacci", seq: anynacci(3)},
s${name: "Tetranacci", seq: anynacci(4)},
s${name: "Lucas", seq: sequence[int]$[2,1]}
]
for seq: s in array[s]$elements(seqs) do
stream$putleft(po, seq.name, 12)
print_n(seq.seq, 10)
end
end start_up- Output:
Fibonacci 1 1 2 3 5 8 13 21 34 55 Tribonacci 1 1 2 4 7 13 24 44 81 149 Tetranacci 1 1 2 4 8 15 29 56 108 208 Lucas 2 1 3 4 7 11 18 29 47 76
Common Lisp
(defun gen-fib (lst m)
"Return the first m members of a generalized Fibonacci sequence using lst as initial values
and the length of lst as step."
(let ((l (- (length lst) 1)))
(do* ((fib-list (reverse lst) (cons (loop for i from 0 to l sum (nth i fib-list)) fib-list))
(c (+ l 2) (+ c 1)))
((> c m) (reverse fib-list)))))
(defun initial-values (n)
"Return the initial values of the Fibonacci n-step sequence"
(cons 1
(loop for i from 0 to (- n 2)
collect (expt 2 i))))
(defun start ()
(format t "Lucas series: ~a~%" (gen-fib '(2 1) 10))
(loop for i from 2 to 4
do (format t "Fibonacci ~a-step sequence: ~a~%" i (gen-fib (initial-values i) 10))))
- Output:
Lucas series: (2 1 3 4 7 11 18 29 47 76) Fibonacci 2-step sequence: (1 1 2 3 5 8 13 21 34 55) Fibonacci 3-step sequence: (1 1 2 4 7 13 24 44 81 149) Fibonacci 4-step sequence: (1 1 2 4 8 15 29 56 108 208)
D
Basic Memoization
void main() {
import std.stdio, std.algorithm, std.range, std.conv;
const(int)[] memo;
size_t addNum;
void setHead(int[] head) nothrow @safe {
memo = head;
addNum = head.length;
}
int fibber(in size_t n) nothrow @safe {
if (n >= memo.length)
memo ~= iota(n - addNum, n).map!fibber.sum;
return memo[n];
}
setHead([1, 1]);
10.iota.map!fibber.writeln;
setHead([2, 1]);
10.iota.map!fibber.writeln;
const prefixes = "fibo tribo tetra penta hexa hepta octo nona deca";
foreach (immutable n, const name; prefixes.split.enumerate(2)) {
setHead(1 ~ iota(n - 1).map!q{2 ^^ a}.array);
writefln("n=%2d, %5snacci -> %(%d %) ...", n, name,
15.iota.map!fibber);
}
}
- Output:
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55] [2, 1, 3, 4, 7, 11, 18, 29, 47, 76] n= 2, fibonacci -> 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ... n= 3, tribonacci -> 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ... n= 4, tetranacci -> 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ... n= 5, pentanacci -> 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ... n= 6, hexanacci -> 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ... n= 7, heptanacci -> 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ... n= 8, octonacci -> 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ... n= 9, nonanacci -> 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ... n=10, decanacci -> 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
Callable Struct
The output is similar.
import std.stdio, std.algorithm, std.range, std.conv;
struct fiblike(T) {
const(T)[] memo;
immutable size_t addNum;
this(in T[] start) nothrow @safe {
this.memo = start.dup;
this.addNum = start.length;
}
T opCall(in size_t n) nothrow @safe {
if (n >= memo.length)
memo ~= iota(n - addNum, n)
.map!(i => opCall(i))
.sum
.to!int;
return memo[n];
}
}
void main() {
auto fibo = fiblike!int([1, 1]);
iota(10).map!fibo.writeln;
auto lucas = fiblike!int([2, 1]);
iota(10).map!lucas.writeln;
const prefixes = "fibo tribo tetra penta hexa hepta octo nona deca";
foreach (immutable n, const name; prefixes.split.enumerate(2)) {
auto fib = fiblike!int(1 ~ iota(n - 1).map!q{2 ^^ a}.array);
writefln("n=%2d, %5snacci -> %(%d %) ...",
n, name, 15.iota.map!fib);
}
}
Struct With opApply
The output is similar.
import std.stdio, std.algorithm, std.range, std.traits;
struct Fiblike(T) {
T[] tail;
int opApply(int delegate(immutable ref T) dg) {
int result, pos;
foreach (immutable x; tail) {
result = dg(x);
if (result)
return result;
}
foreach (immutable i; tail.length.iota.cycle) {
immutable x = tail.sum;
result = dg(x);
if (result)
break;
tail[i] = x;
}
return result;
}
}
// std.range.take doesn't work with opApply.
ForeachType!It[] takeApply(It)(It iterable, in size_t n) {
typeof(return) result;
foreach (immutable x; iterable) {
result ~= x;
if (result.length == n)
break;
}
return result;
}
void main() {
Fiblike!int([1, 1]).takeApply(10).writeln;
Fiblike!int([2, 1]).takeApply(10).writeln;
const prefixes = "fibo tribo tetra penta hexa hepta octo nona deca";
foreach (immutable n, const name; prefixes.split.enumerate(2)) {
auto fib = Fiblike!int(1 ~ iota(n - 1).map!q{2 ^^ a}.array);
writefln("n=%2d, %5snacci -> %s", n, name, fib.takeApply(15));
}
}
Range Generator Version
void main() {
import std.stdio, std.algorithm, std.range, std.concurrency;
immutable fibLike = (int[] tail) => new Generator!int({
foreach (x; tail)
yield(x);
foreach (immutable i; tail.length.iota.cycle)
yield(tail[i] = tail.sum);
});
foreach (seed; [[1, 1], [2, 1]])
fibLike(seed).take(10).writeln;
immutable prefixes = "fibo tribo tetra penta hexa hepta octo nona deca";
foreach (immutable n, const name; prefixes.split.enumerate(2)) {
auto fib = fibLike(1 ~ iota(n - 1).map!q{2 ^^ a}.array);
writefln("n=%2d, %5snacci -> %(%s, %), ...", n, name, fib.take(15));
}
}
- Output:
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55] [2, 1, 3, 4, 7, 11, 18, 29, 47, 76] n= 2, fibonacci -> 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ... n= 3, tribonacci -> 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, ... n= 4, tetranacci -> 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, ... n= 5, pentanacci -> 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, ... n= 6, hexanacci -> 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, ... n= 7, heptanacci -> 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, ... n= 8, octonacci -> 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, ... n= 9, nonanacci -> 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, ... n=10, decanacci -> 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, ...
Delphi
See #Pascal.
EasyLang
proc sequ n$ val[] n . .
write n$ & ": "
il = len val[]
len val[] n
for i = il + 1 to n
for j = 1 to il
val[i] += val[i - j]
.
.
for v in val[]
write v & " "
.
print ""
.
sequ "Fibonacci" [ 1 1 ] 10
sequ "Tribonacci" [ 1 1 2 ] 10
sequ "Tetrabonacci" [ 1 1 2 4 ] 10
sequ "Lucas" [ 2 1 ] 10- Output:
Fibonacci: 1 1 2 3 5 8 13 21 34 55 Tribonacci: 1 1 2 4 7 13 24 44 81 149 Tetrabonacci: 1 1 2 4 8 15 29 56 108 208 Lucas: 2 1 3 4 7 11 18 29 47 76
EchoLisp
;; generate a recursive lambda() for a x-nacci
;; equip it with memoïzation
;; bind it to its name
(define (make-nacci name seed)
(define len (1+ (vector-length seed)))
(define-global name
`(lambda(n) (for/sum ((i (in-range (1- n) (- n ,len) -1))) (,name i))))
(remember name seed)
name)
(define nacci-family `(
(Fibonacci #(1 1))
(Tribonacci #(1 1 2))
(Tetranacci #(1 1 2 4))
(Decanacci #(1 1 2 4 8 16 32 64 128 256))
(Random-😜-nacci ,(list->vector (take 6 (shuffle (iota 100)))))
(Lucas #(2 1))))
(define (task naccis)
(for ((nacci naccis))
(define-values (name seed) nacci)
(make-nacci name seed)
(printf "%s[%d] → %d" name (vector-length seed) (take name 16))))
- Output:
(task nacci-family ) Fibonacci[2] → (1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987) Tribonacci[3] → (1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768) Tetranacci[4] → (1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671) Decanacci[10] → (1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 16336) Random-😜-nacci[6] → (95 52 16 48 59 56 326 557 1062 2108 4168 8277 16498 32670 64783 128504) Lucas[2] → (2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364)
Elixir
defmodule RC do
def anynacci(start_sequence, count) do
n = length(start_sequence)
anynacci(Enum.reverse(start_sequence), count-n, n)
end
def anynacci(seq, 0, _), do: Enum.reverse(seq)
def anynacci(seq, count, n) do
next = Enum.sum(Enum.take(seq, n))
anynacci([next|seq], count-1, n)
end
end
IO.inspect RC.anynacci([1,1], 15)
naccis = [ lucus: [2,1],
fibonacci: [1,1],
tribonacci: [1,1,2],
tetranacci: [1,1,2,4],
pentanacci: [1,1,2,4,8],
hexanacci: [1,1,2,4,8,16],
heptanacci: [1,1,2,4,8,16,32],
octonacci: [1,1,2,4,8,16,32,64],
nonanacci: [1,1,2,4,8,16,32,64,128],
decanacci: [1,1,2,4,8,16,32,64,128,256] ]
Enum.each(naccis, fn {name, list} ->
:io.format("~11s: ", [name])
IO.inspect RC.anynacci(list, 15)
end)
- Output:
lucus: [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843] fibonacci: [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610] tribonacci: [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136] tetranacci: [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536] pentanacci: [1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930] hexanacci: [1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617] heptanacci: [1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936] octonacci: [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080] nonanacci: [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144] decanacci: [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172]
Erlang
-module( fibonacci_nstep ).
-export( [nacci/2, task/0] ).
nacci( N, Ns ) when N =< erlang:length(Ns) ->
{Sequence, _Not_sequence} = lists:split( N, Ns ),
Sequence;
nacci( N, Ns ) ->
Nth = erlang:length( Ns ),
{_Nth, Sequence_reversed} = lists:foldl( fun nacci_foldl/2, {Nth, lists:reverse(Ns)}, lists:seq(Nth+1, N) ),
lists:reverse( Sequence_reversed ).
task() ->
Names_and_funs = [{X, fun (N) -> nacci( N, Y ) end} || {X, Y} <- [{fibonacci, [1, 1]}, {tribonacci, [1, 1, 2]}, {tetranacci, [1, 1, 2, 4]}, {lukas, [2, 1]}]],
[io:fwrite( "~p: ~p~n", [X, Y(10)] ) || {X, Y} <- Names_and_funs].
nacci_foldl( _N, {Nth, Ns} ) ->
{Sum_ns, _Not_sum_ns} = lists:split( Nth, Ns ),
{Nth, [lists:sum(Sum_ns) | Ns]}.
- Output:
59> fibonacci_nstep:task(). fibonacci: [1,1,2,3,5,8,13,21,34,55] tribonacci: [1,1,2,4,7,13,24,44,81,149] tetranacci: [1,1,2,4,8,15,29,56,108,208] lukas: [2,1,3,4,7,11,18,29,47,76]
ERRE
PROGRAM FIBON
!
! for rosettacode.org
!
DIM F[20]
PROCEDURE FIB(TIPO$,F$)
FOR I%=0 TO 20 DO
F[I%]=0
END FOR
B=0
LOOP
Q=INSTR(F$,",")
B=B+1
IF Q=0 THEN
F[B]=VAL(F$)
EXIT
ELSE
F[B]=VAL(MID$(F$,1,Q-1))
F$=MID$(F$,Q+1)
END IF
END LOOP
PRINT(TIPO$;" =>";)
FOR I=B TO 14+B DO
IF I<>B THEN PRINT(",";) END IF
PRINT(F[I-B+1];)
FOR J=(I-B)+1 TO I DO
F[I+1]=F[I+1]+F[J]
END FOR
END FOR
PRINT
END PROCEDURE
BEGIN
PRINT(CHR$(12);) ! CLS
FIB("Fibonacci","1,1")
FIB("Tribonacci","1,1,2")
FIB("Tetranacci","1,1,2,4")
FIB("Lucas","2,1")
END PROGRAMF#
let fibinit = Seq.append (Seq.singleton 1) (Seq.unfold (fun n -> Some(n, 2*n)) 1)
let fiblike init =
Seq.append
(Seq.ofList init)
(Seq.unfold
(function | least :: rest ->
let this = least + Seq.reduce (+) rest
Some(this, rest @ [this])
| _ -> None) init)
let lucas = fiblike [2; 1]
let nacci n = Seq.take n fibinit |> Seq.toList |> fiblike
[<EntryPoint>]
let main argv =
let start s = Seq.take 15 s |> Seq.toList
let prefix = "fibo tribo tetra penta hexa hepta octo nona deca".Split()
Seq.iter
(fun (p, n) -> printfn "n=%2i, %5snacci -> %A" n p (start (nacci n)))
(Seq.init prefix.Length (fun i -> (prefix.[i], i+2)))
printfn " lucas -> %A" (start (fiblike [2; 1]))
0
Output
n= 2, fibonacci -> [1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377; 610]
n= 3, tribonacci -> [1; 1; 2; 4; 7; 13; 24; 44; 81; 149; 274; 504; 927; 1705; 3136]
n= 4, tetranacci -> [1; 1; 2; 4; 8; 15; 29; 56; 108; 208; 401; 773; 1490; 2872; 5536]
n= 5, pentanacci -> [1; 1; 2; 4; 8; 16; 31; 61; 120; 236; 464; 912; 1793; 3525; 6930]
n= 6, hexanacci -> [1; 1; 2; 4; 8; 16; 32; 63; 125; 248; 492; 976; 1936; 3840; 7617]
n= 7, heptanacci -> [1; 1; 2; 4; 8; 16; 32; 64; 127; 253; 504; 1004; 2000; 3984; 7936]
n= 8, octonacci -> [1; 1; 2; 4; 8; 16; 32; 64; 128; 255; 509; 1016; 2028; 4048; 8080]
n= 9, nonanacci -> [1; 1; 2; 4; 8; 16; 32; 64; 128; 256; 511; 1021; 2040; 4076; 8144]
n=10, decanacci -> [1; 1; 2; 4; 8; 16; 32; 64; 128; 256; 512; 1023; 2045; 4088; 8172]
lucas -> [2; 1; 3; 4; 7; 11; 18; 29; 47; 76; 123; 199; 322; 521; 843]
Factor
building is a dynamic variable that refers to the sequence being built by make. This is useful when the next element of the sequence depends on previous elements.
USING: formatting fry kernel make math namespaces qw sequences ;
: n-bonacci ( n initial -- seq ) [
[ [ , ] each ] [ length - ] [ length ] tri
'[ building get _ tail* sum , ] times
] { } make ;
qw{ fibonacci tribonacci tetranacci lucas }
{ { 1 1 } { 1 1 2 } { 1 1 2 4 } { 2 1 } }
[ 10 swap n-bonacci "%-10s %[%3d, %]\n" printf ] 2each
- Output:
fibonacci { 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 }
tribonacci { 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 }
tetranacci { 1, 1, 2, 4, 8, 15, 29, 56, 108, 208 }
lucas { 2, 1, 3, 4, 7, 11, 18, 29, 47, 76 }
Forth
: length @ ; \ length of an array is stored at its address
: a{ here cell allot ;
: } , here over - cell / over ! ;
defer nacci
: step ( a- i n -- a- i m )
>r 1- 2dup nacci r> + ;
: steps ( a- i n -- m )
0 tuck do step loop nip nip ;
:noname ( a- i -- n )
over length over > \ if i is within the array
if cells + @ \ fetch i...if not,
else over length 1- steps \ get length of array for calling step and recurse
then ; is nacci
: show-nacci 11 1 do dup i nacci . loop cr drop ;
." fibonacci: " a{ 1 , 1 } show-nacci
." tribonacci: " a{ 1 , 1 , 2 } show-nacci
." tetranacci: " a{ 1 , 1 , 2 , 4 } show-nacci
." lucas: " a{ 2 , 1 } show-nacci
- Output:
fibonacci: 1 1 2 3 5 8 13 21 34 55 tribonacci: 1 1 2 4 7 13 24 44 81 149 tetranacci: 1 1 2 4 8 15 29 56 108 208 lucas: 2 1 3 4 7 11 18 29 47 76
Fortran
! save this program as file f.f08
! gnu-linux command to build and test
! $ a=./f && gfortran -Wall -std=f2008 $a.f08 -o $a && echo -e 2\\n5\\n\\n | $a
! -*- mode: compilation; default-directory: "/tmp/" -*-
! Compilation started at Fri Apr 4 23:20:27
!
! a=./f && gfortran -Wall -std=f2008 $a.f08 -o $a && echo -e 2\\n8\\ny\\n | $a
! Enter the number of terms to sum: Show the the first how many terms of the sequence? Accept this initial sequence (y/n)?
! 1 1
! 1 1 2 3 5 8 13 21
!
! Compilation finished at Fri Apr 4 23:20:27
program f
implicit none
integer :: n, terms
integer, allocatable, dimension(:) :: sequence
integer :: i
character :: answer
write(6,'(a)',advance='no')'Enter the number of terms to sum: '
read(5,*) n
if ((n < 2) .or. (29 < n)) stop'Unreasonable! Exit.'
write(6,'(a)',advance='no')'Show the the first how many terms of the sequence? '
read(5,*) terms
if (terms < 1) stop'Lazy programmer has not implemented backward sequences.'
n = min(n, terms)
allocate(sequence(1:terms))
sequence(1) = 1
do i = 0, n - 2
sequence(i+2) = 2**i
end do
write(6,*)'Accept this initial sequence (y/n)?'
write(6,*) sequence(:n)
read(5,*) answer
if (answer .eq. 'n') then
write(6,*) 'Fine. Enter the initial terms.'
do i=1, n
write(6, '(i2,a2)', advance = 'no') i, ': '
read(5, *) sequence(i)
end do
end if
call nacci(n, sequence)
write(6,*) sequence(:terms)
deallocate(sequence)
contains
subroutine nacci(n, s)
! nacci =: (] , +/@{.)^:(-@#@]`(-#)`])
integer, intent(in) :: n
integer, intent(inout), dimension(:) :: s
integer :: i, terms
terms = size(s)
! do i = n+1, terms
! s(i) = sum(s(i-n:i-1))
! end do
i = n+1
if (n+1 .le. terms) s(i) = sum(s(i-n:i-1))
do i = n + 2, terms
s(i) = 2*s(i-1) - s(i-(n+1))
end do
end subroutine nacci
end program f
$ ./f # Lucas series
Enter the number of terms to sum: 2
Show the the first how many terms of the sequence? 10
Accept this initial sequence (y/n)?
1 1
n
Fine. Enter the initial terms.
1: 2
2: 1
2 1 3 4 7 11 18 29 47 76
$
$
$
$
$
$
$ ./f # Waltzing the 6-step
Enter the number of terms to sum: 6
Show the the first how many terms of the sequence? 10
Accept this initial sequence (y/n)?
1 1 2 4 8 16
y
1 1 2 4 8 16 32 63 125 248
$
$ for n in 2 3 4;do echo -e $n\\n10\\ny|./f;done
Enter the number of terms to sum: Show the the first how many terms of the sequence? Accept this initial sequence (y/n)?
1 1
1 1 2 3 5 8 13 21 34 55
Enter the number of terms to sum: Show the the first how many terms of the sequence? Accept this initial sequence (y/n)?
1 1 2
1 1 2 4 7 13 24 44 81 149
Enter the number of terms to sum: Show the the first how many terms of the sequence? Accept this initial sequence (y/n)?
1 1 2 4
1 1 2 4 8 15 29 56 108 208
FreeBASIC
' FB 1.05.0 Win64
' Deduces the step, n, from the length of the dynamic array passed in
' and fills it out to 'size' elements
Sub fibN (a() As Integer, size As Integer)
Dim lb As Integer = LBound(a)
Dim ub As Integer = UBound(a)
Dim length As Integer = ub - lb + 1
If length < 2 OrElse length >= size Then Return
ub = lb + size - 1
Redim Preserve a(lb To ub)
Dim sum As Integer
For i As Integer = lb + length to ub
sum = 0
For j As Integer = 1 To Length
sum += a(i - j)
Next j
a(i) = sum
Next i
End Sub
Sub printSeries(a() As Integer, name_ As String) '' name is a keyword
Print name_; " =>";
For i As Integer = LBound(a) To UBound(a)
Print Using "####"; a(i);
Print " ";
Next
Print
End Sub
Const size As Integer = 13 '' say
Redim a(1 To 2) As Integer
a(1) = 1 : a(2) = 1
fibN(a(), size)
printSeries(a(), "fibonacci ")
Redim Preserve a(1 To 3)
a(3) = 2
fibN(a(), size)
printSeries(a(), "tribonacci")
Redim Preserve a(1 To 4)
a(4) = 4
fibN(a(), size)
printSeries(a(), "tetranacci")
erase a
Redim a(1 To 2)
a(1) = 2 : a(2) = 1
fibN(a(), size)
printSeries(a(), "lucas ")
Print
Print "Press any key to quit"
Sleep
- Output:
fibonacci => 1 1 2 3 5 8 13 21 34 55 89 144 233 tribonacci => 1 1 2 4 7 13 24 44 81 149 274 504 927 tetranacci => 1 1 2 4 8 15 29 56 108 208 401 773 1490 lucas => 2 1 3 4 7 11 18 29 47 76 123 199 322
FunL
import util.TextTable
native scala.collection.mutable.Queue
def fibLike( init ) =
q = Queue()
for i <- init do q.enqueue( i )
def fib =
q.enqueue( sum(q) )
q.dequeue() # fib()
0 # fib()
def fibN( n ) = fibLike( [1] + [2^i | i <- 0:n-1] )
val lucas = fibLike( [2, 1] )
t = TextTable()
t.header( 'k', 'Fibonacci', 'Tribonacci', 'Tetranacci', 'Lucas' )
t.line()
for i <- 1..5
t.rightAlignment( i )
seqs = (fibN(2), fibN(3), fibN(4), lucas)
for k <- 1..10
t.row( ([k] + [seqs(i)(k) | i <- 0:4]).toIndexedSeq() )
print( t )- Output:
+----+-----------+------------+------------+-------+ | k | Fibonacci | Tribonacci | Tetranacci | Lucas | +----+-----------+------------+------------+-------+ | 1 | 1 | 1 | 1 | 2 | | 2 | 1 | 1 | 1 | 1 | | 3 | 2 | 2 | 2 | 3 | | 4 | 3 | 4 | 4 | 4 | | 5 | 5 | 7 | 8 | 7 | | 6 | 8 | 13 | 15 | 11 | | 7 | 13 | 24 | 29 | 18 | | 8 | 21 | 44 | 56 | 29 | | 9 | 34 | 81 | 108 | 47 | | 10 | 55 | 149 | 208 | 76 | +----+-----------+------------+------------+-------+
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
According to the requirements, the program must generate a series, and the order (Fibonacci, Tribonacci, etc) should be determined according with the initial values.
In this case, the number n indicates how many terms of the series will be generated.
The following generates a Fibonacci series of 15 terms:
The following generates a Lucas series of 15 terms:
The following generates a Tribonacci series of 15 terms:
Generating initial values. The initial values can be generated by the following function:
Note that it is a recursive function, and it calls the previously defined function. It requires the initial values as a seed: (1, 1) for Fibonacci style (Fibonacci, Tribonacci, etc), and (2, 1) for Lucas style.
The following generates the initial values for Fibonacci series.
The following generates the initial values for Lucas series.
Generating tables of series for Fibonacci and Lucas
This generates a tables of series for Fibonacci (15 terms), for orders 2 to 10 (Fibonacci, Tribonacci, etc.)
This generates a tables of series for Lucas (15 terms), for orders 2 to 15:
Go
Solution using separate goroutines.
package main
import "fmt"
func g(i []int, c chan<- int) {
var sum int
b := append([]int(nil), i...) // make a copy
for _, t := range b {
c <- t
sum += t
}
for {
for j, t := range b {
c <- sum
b[j], sum = sum, sum+sum-t
}
}
}
func main() {
for _, s := range [...]struct {
seq string
i []int
}{
{"Fibonacci", []int{1, 1}},
{"Tribonacci", []int{1, 1, 2}},
{"Tetranacci", []int{1, 1, 2, 4}},
{"Lucas", []int{2, 1}},
} {
fmt.Printf("%10s:", s.seq)
c := make(chan int)
// Note/warning: these goroutines are leaked.
go g(s.i, c)
for j := 0; j < 10; j++ {
fmt.Print(" ", <-c)
}
fmt.Println()
}
}
- Output:
Fibonacci: 1 1 2 3 5 8 13 21 34 55
Tribonacci: 1 1 2 4 7 13 24 44 81 149
Tetranacci: 1 1 2 4 8 15 29 56 108 208
Lucas: 2 1 3 4 7 11 18 29 47 76
Groovy
Solution
def fib = { List seed, int k=10 ->
assert seed : "The seed list must be non-null and non-empty"
assert seed.every { it instanceof Number } : "Every member of the seed must be a number"
def n = seed.size()
assert n > 1 : "The seed must contain at least two elements"
List result = [] + seed
if (k < n) {
result[0..k]
} else {
(n..k).inject(result) { res, kk ->
res << res[-n..-1].sum()
}
}
}
Test
[
' fibonacci':[1,1],
'tribonacci':[1,1,2],
'tetranacci':[1,1,2,4],
'pentanacci':[1,1,2,4,8],
' hexanacci':[1,1,2,4,8,16],
'heptanacci':[1,1,2,4,8,16,32],
' octonacci':[1,1,2,4,8,16,32,64],
' nonanacci':[1,1,2,4,8,16,32,64,128],
' decanacci':[1,1,2,4,8,16,32,64,128,256],
' lucas':[2,1],
].each { name, seed ->
println "${name}: ${fib(seed,10)}"
}
println " lucas[0]: ${fib([2,1],0)}"
println " tetra[3]: ${fib([1,1,2,4],3)}"
- Output:
fibonacci: [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
tribonacci: [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274]
tetranacci: [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401]
pentanacci: [1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464]
hexanacci: [1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492]
heptanacci: [1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504]
octonacci: [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509]
nonanacci: [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511]
decanacci: [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
lucas: [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
lucas[0]: [2]
tetra[3]: [1, 1, 2, 4]
Haskell
import Control.Monad (zipWithM_)
import Data.List (tails)
fiblike :: [Integer] -> [Integer]
fiblike st = xs
where
xs = st <> map (sum . take n) (tails xs)
n = length st
nstep :: Int -> [Integer]
nstep n = fiblike $ take n $ 1 : iterate (2 *) 1
main :: IO ()
main = do
mapM_ (print . take 10 . fiblike) [[1, 1], [2, 1]]
zipWithM_
( \n name -> do
putStr (name <> "nacci -> ")
print $ take 15 $ nstep n
)
[2 ..]
(words "fibo tribo tetra penta hexa hepta octo nona deca")
- Output:
[1,1,2,3,5,8,13,21,34,55] [2,1,3,4,7,11,18,29,47,76] fibonacci -> [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610] tribonacci -> [1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136] tetranacci -> [1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536] pentanacci -> [1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930] hexanacci -> [1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617] heptanacci -> [1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936] octonacci -> [1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080] nonanacci -> [1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144] decanacci -> [1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172]
Or alternatively, without imports – using only the default Prelude:
------------ FIBONACCI N-STEP NUMBER SEQUENCES -----------
nStepFibonacci :: Int -> [Int]
nStepFibonacci =
nFibs
. (1 :)
. fmap (2 ^)
. enumFromTo 0
. subtract 2
nFibs :: [Int] -> [Int]
nFibs ys@(z : zs) = z : nFibs (zs <> [sum ys])
--------------------------- TEST -------------------------
main :: IO ()
main = do
putStrLn $
justifyLeft 12 ' ' "Lucas" <> "-> "
<> show (take 15 (nFibs [2, 1]))
(putStrLn . unlines)
( zipWith
( \s n ->
justifyLeft 12 ' ' (s <> "naccci")
<> ("-> " <> show (take 15 (nStepFibonacci n)))
)
( words
"fibo tribo tetra penta hexa hepta octo nona deca"
)
[2 ..]
)
justifyLeft :: Int -> Char -> String -> String
justifyLeft n c s = take n (s <> replicate n c)
Lucas -> [2,1,3,4,7,11,18,29,47,76,123,199,322,521,843] fibonaccci -> [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610] tribonaccci -> [1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136] tetranaccci -> [1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536] pentanaccci -> [1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930] hexanaccci -> [1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617] heptanaccci -> [1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936] octonaccci -> [1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080] nonanaccci -> [1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144] decanaccci -> [1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172]
or in terms of unfoldr:
import Data.Bifunctor (second)
import Data.List (transpose, uncons, unfoldr)
------------ FIBONACCI N-STEP NUMBER SEQUENCES -----------
a000032 :: [Int]
a000032 = unfoldr (recurrence 2) [2, 1]
nStepFibonacci :: Int -> [Int]
nStepFibonacci =
unfoldr <$> recurrence
<*> (($ 1 : fmap (2 ^) [0 ..]) . take)
recurrence :: Int -> [Int] -> Maybe (Int, [Int])
recurrence n =
( fmap
. second
. flip (<>)
. pure
. sum
. take n
)
<*> uncons
--------------------------- TEST -------------------------
main :: IO ()
main =
putStrLn $
"Recurrence relation sequences:\n\n"
<> spacedTable
justifyRight
( ("lucas:" : fmap show (take 15 a000032)) :
zipWith
( \k n ->
(k <> "nacci:") :
fmap
show
(take 15 $ nStepFibonacci n)
)
(words "fibo tribo tetra penta hexa hepta octo nona deca")
[2 ..]
)
------------------------ FORMATTING ----------------------
spacedTable ::
(Int -> Char -> String -> String) -> [[String]] -> String
spacedTable aligned rows =
let columnWidths =
fmap
(maximum . fmap length)
(transpose rows)
in unlines $
fmap
(unwords . zipWith (`aligned` ' ') columnWidths)
rows
justifyRight :: Int -> Char -> String -> String
justifyRight n c = (drop . length) <*> (replicate n c <>)
- Output:
Recurrence relation sequences:
lucas: 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843
fibonacci: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
tribonacci: 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136
tetranacci: 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536
pentanacci: 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930
hexanacci: 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617
heptanacci: 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936
octonacci: 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080
nonanacci: 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144
decanacci: 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172
Icon and Unicon
Works in both languages:
procedure main(A)
every writes("F2:\t"|right((fnsGen(1,1))\14,5) | "\n")
every writes("F3:\t"|right((fnsGen(1,1,2))\14,5) | "\n")
every writes("F4:\t"|right((fnsGen(1,1,2,4))\14,5) | "\n")
every writes("Lucas:\t"|right((fnsGen(2,1))\14,5) | "\n")
every writes("F?:\t"|right((fnsGen!A)\14,5) | "\n")
end
procedure fnsGen(cache[])
n := *cache
every i := seq() do {
if i > *cache then every (put(cache,0),cache[i] +:= cache[i-n to i-1])
suspend cache[i]
}
end
Output:
->fns 3 1 4 1 5 F2: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 F3: 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 F4: 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 Lucas: 2 1 3 4 7 11 18 29 47 76 123 199 322 521 F?: 3 1 4 1 5 14 25 49 94 187 369 724 1423 2797 ->
A slightly longer version of fnsGen that reduces the memory footprint is:
procedure fnsGen(cache[])
every i := seq() do {
if i := (i > *cache, *cache) then {
every (sum := 0) +:= !cache
put(cache, sum) # cache only 'just enough'
pop(cache)
}
suspend cache[i]
}
end
The output is identical.
J
Solution:
nacci =: (] , +/@{.)^:(-@#@]`(-#)`])
Example (Lucas):
10 nacci 2 1 NB. Lucas series, first 10 terms
2 1 3 4 7 11 18 29 47 76
Example (extended 'nacci series):
TESTS =: }."1 fixdsv noun define [ require 'tables/dsv' NB. Tests from task description
2 fibonacci 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...
3 tribonacci 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ...
4 tetranacci 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ...
5 pentanacci 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ...
6 hexanacci 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ...
7 heptanacci 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ...
8 octonacci 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ...
9 nonanacci 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ...
10 decanacci 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
)
testNacci =: ] -: #@] nacci {. NB. Given an order & test sequence, compare nacci to sequence
OT =: __ ".&.> (<<<1) { |: TESTS NB. 'nacci order and test sequence
(> 1 {"1 TESTS) ,. ' ' ,. (u: 16b274c 16b2713) {~ (testNacci }:)&>/ OT NB. ✓ or ❌ for success or failure
fibonacci ✓
tribonacci ✓
tetranacci ✓
pentanacci ✓
hexanacci ✓
heptanacci ✓
octonacci ✓
nonanacci ✓
decanacci ✓
Java
Code:
class Fibonacci
{
public static int[] lucas(int n, int numRequested)
{
if (n < 2)
throw new IllegalArgumentException("Fibonacci value must be at least 2");
return fibonacci((n == 2) ? new int[] { 2, 1 } : lucas(n - 1, n), numRequested);
}
public static int[] fibonacci(int n, int numRequested)
{
if (n < 2)
throw new IllegalArgumentException("Fibonacci value must be at least 2");
return fibonacci((n == 2) ? new int[] { 1, 1 } : fibonacci(n - 1, n), numRequested);
}
public static int[] fibonacci(int[] startingValues, int numRequested)
{
int[] output = new int[numRequested];
int n = startingValues.length;
System.arraycopy(startingValues, 0, output, 0, n);
for (int i = n; i < numRequested; i++)
for (int j = 1; j <= n; j++)
output[i] += output[i - j];
return output;
}
public static void main(String[] args)
{
for (int n = 2; n <= 10; n++)
{
System.out.print("nacci(" + n + "):");
for (int value : fibonacci(n, 15))
System.out.print(" " + value);
System.out.println();
}
for (int n = 2; n <= 10; n++)
{
System.out.print("lucas(" + n + "):");
for (int value : lucas(n, 15))
System.out.print(" " + value);
System.out.println();
}
}
}
Output:
nacci(2): 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 nacci(3): 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 nacci(4): 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 nacci(5): 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 nacci(6): 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 nacci(7): 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 nacci(8): 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 nacci(9): 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 nacci(10): 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 lucas(2): 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 lucas(3): 2 1 3 6 10 19 35 64 118 217 399 734 1350 2483 4567 lucas(4): 2 1 3 6 12 22 43 83 160 308 594 1145 2207 4254 8200 lucas(5): 2 1 3 6 12 24 46 91 179 352 692 1360 2674 5257 10335 lucas(6): 2 1 3 6 12 24 48 94 187 371 736 1460 2896 5744 11394 lucas(7): 2 1 3 6 12 24 48 96 190 379 755 1504 2996 5968 11888 lucas(8): 2 1 3 6 12 24 48 96 192 382 763 1523 3040 6068 12112 lucas(9): 2 1 3 6 12 24 48 96 192 384 766 1531 3059 6112 12212 lucas(10): 2 1 3 6 12 24 48 96 192 384 768 1534 3067 6131 12256
JavaScript
ES5
function fib(arity, len) {
return nacci(nacci([1,1], arity, arity), arity, len);
}
function lucas(arity, len) {
return nacci(nacci([2,1], arity, arity), arity, len);
}
function nacci(a, arity, len) {
while (a.length < len) {
var sum = 0;
for (var i = Math.max(0, a.length - arity); i < a.length; i++)
sum += a[i];
a.push(sum);
}
return a;
}
function main() {
for (var arity = 2; arity <= 10; arity++)
console.log("fib(" + arity + "): " + fib(arity, 15));
for (var arity = 2; arity <= 10; arity++)
console.log("lucas(" + arity + "): " + lucas(arity, 15));
}
main();
- Output:
fib(2): 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610 fib(3): 1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136 fib(4): 1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536 fib(5): 1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930 fib(6): 1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617 fib(7): 1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936 fib(8): 1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080 fib(9): 1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144 fib(10): 1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172 lucas(2): 2,1,3,4,7,11,18,29,47,76,123,199,322,521,843 lucas(3): 2,1,3,6,10,19,35,64,118,217,399,734,1350,2483,4567 lucas(4): 2,1,3,6,12,22,43,83,160,308,594,1145,2207,4254,8200 lucas(5): 2,1,3,6,12,24,46,91,179,352,692,1360,2674,5257,10335 lucas(6): 2,1,3,6,12,24,48,94,187,371,736,1460,2896,5744,11394 lucas(7): 2,1,3,6,12,24,48,96,190,379,755,1504,2996,5968,11888 lucas(8): 2,1,3,6,12,24,48,96,192,382,763,1523,3040,6068,12112 lucas(9): 2,1,3,6,12,24,48,96,192,384,766,1531,3059,6112,12212 lucas(10): 2,1,3,6,12,24,48,96,192,384,768,1534,3067,6131,12256
ES6
(() => {
'use strict';
// Start sequence -> Number of terms -> terms
// takeNFibs :: [Int] -> Int -> [Int]
const takeNFibs = (xs, n) => {
const go = (xs, n) =>
0 < n && 0 < xs.length ? (
cons(
head(xs),
go(
append(tail(xs), [sum(xs)]),
n - 1
)
)
) : [];
return go(xs, n);
};
// fibInit :: Int -> [Int]
const fibInit = n =>
cons(
1,
map(x => Math.pow(2, x),
enumFromToInt(0, n - 2)
)
);
// TEST -----------------------------------------------------------------
const main = () => {
const
intTerms = 15,
strTable = unlines(
zipWith(
(s, n) =>
justifyLeft(12, ' ', s + 'nacci') + ' -> ' +
showJSON(
takeNFibs(fibInit(n), intTerms)
),
words('fibo tribo tetra penta hexa hepta octo nona deca'),
enumFromToInt(2, 10)
)
);
return justifyLeft(12, ' ', 'Lucas ') + ' -> ' +
showJSON(takeNFibs([2, 1], intTerms)) + '\n' +
strTable;
};
// GENERIC FUNCTIONS ----------------------------
// append (++) :: [a] -> [a] -> [a]
// append (++) :: String -> String -> String
const append = (xs, ys) => xs.concat(ys);
// cons :: a -> [a] -> [a]
const cons = (x, xs) =>
Array.isArray(xs) ? (
[x].concat(xs)
) : (x + xs);
// enumFromToInt :: Int -> Int -> [Int]
const enumFromToInt = (m, n) =>
m <= n ? iterateUntil(
x => n <= x,
x => 1 + x,
m
) : [];
// head :: [a] -> a
const head = xs => xs.length ? xs[0] : undefined;
// iterateUntil :: (a -> Bool) -> (a -> a) -> a -> [a]
const iterateUntil = (p, f, x) => {
const vs = [x];
let h = x;
while (!p(h))(h = f(h), vs.push(h));
return vs;
};
// justifyLeft :: Int -> Char -> String -> String
const justifyLeft = (n, cFiller, s) =>
n > s.length ? (
s.padEnd(n, cFiller)
) : s;
// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) => xs.map(f);
// showJSON :: a -> String
const showJSON = x => JSON.stringify(x);
// sum :: [Num] -> Num
const sum = xs => xs.reduce((a, x) => a + x, 0);
// tail :: [a] -> [a]
const tail = xs => 0 < xs.length ? xs.slice(1) : [];
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// words :: String -> [String]
const words = s => s.split(/\s+/);
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = (f, xs, ys) =>
Array.from({
length: Math.min(xs.length, ys.length)
}, (_, i) => f(xs[i], ys[i], i));
// MAIN ---
return main();
})();
Lucas -> [2,1,3,4,7,11,18,29,47,76,123,199,322,521,843] fibonacci -> [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610] tribonacci -> [1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136] tetranacci -> [1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536] pentanacci -> [1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930] hexanacci -> [1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617] heptanacci -> [1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936] octonacci -> [1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080] nonanacci -> [1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144] decanacci -> [1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172]
jq
# Input: the initial array
def nacci(arity; len):
arity as $arity | len as $len
| reduce range(length; $len) as $i
(.;
([0, (length - $arity)] | max ) as $lower
| . + [ .[ ($lower) : length] | add] ) ;
def fib(arity; len):
arity as $arity | len as $len
| [1,1] | nacci($arity; $arity) | nacci($arity; $len) ;
def lucas(arity; len):
arity as $arity | len as $len
| [2,1] | nacci($arity; $arity) | nacci($arity; $len) ;Example:
def main:
(range(2; 11) | "fib(\(.)): \(fib(.; 15))"),
(range(2; 11) | "lucas(\(.)): \(lucas(.; 15))")
;
main- Output:
$ jq -M -r -n -f fibonacci_n-step.jq ... [as for JavaScript] ...
Julia
This solution provides a generalized Fibonacci iterator that is then made specific to particular sorts of series by setting its parameters. NFib is the type that holds the series parameters. FState contains the iteration state. The methods start, end and next, provided for these new types, enable Julia's iteration mechanics upon them.
This iterator is implemented using an n-element circular list that contains the previous values of the sequence that are needed to calculate the current value. To do this without clumsy initialization logic, the "seed" sequence consists of the values prior to rather than the first values. For example the (2 step) Fibonacci sequence is with and rather than and . See Primes in Fibonacci n-step and Lucas n-step Sequences for further details.
Generalized Fibonacci Iterator Definition
type NFib{T<:Integer}
n::T
klim::T
seeder::Function
end
type FState
a::Array{BigInt,1}
adex::Integer
k::Integer
end
function Base.start{T<:Integer}(nf::NFib{T})
a = nf.seeder(nf.n)
adex = 1
k = 1
return FState(a, adex, k)
end
function Base.done{T<:Integer}(nf::NFib{T}, fs::FState)
fs.k > nf.klim
end
function Base.next{T<:Integer}(nf::NFib{T}, fs::FState)
f = sum(fs.a)
fs.a[fs.adex] = f
fs.adex = rem1(fs.adex+1, nf.n)
fs.k += 1
return (f, fs)
end
Specification of the n-step Fibonacci Iterator
The seeding for this series of sequences is and .
function fib_seeder{T<:Integer}(n::T)
a = zeros(BigInt, n)
a[1] = one(BigInt)
return a
end
function fib{T<:Integer}(n::T, k::T)
NFib(n, k, fib_seeder)
end
Specification of the Rosetta Code n-step Lucas Iterator
This iterator produces the task description's version of the Lucas Sequence (OEIS A000032) and its generalization to n-steps as was done by some of the other solutions to this task. The seeding for this series of sequences is , and, for , .
function luc_rc_seeder{T<:Integer}(n::T)
a = zeros(BigInt, n)
a[1] = 3
a[2] = -1
return a
end
function luc_rc{T<:Integer}(n::T, k::T)
NFib(n, k, luc_rc_seeder)
end
Specification of the MathWorld n-step Lucas Iterator
This iterator produces the Mathworld version of the Lucas Sequence (Lucas Number and OEIS A000204) and its generalization to n-steps according to Mathworld (Lucas n-Step Number and Primes in Fibonacci n-step and Lucas n-step Sequences). The seeding for this series of sequences is and .
function luc_seeder{T<:Integer}(n::T)
a = -ones(BigInt, n)
a[end] = big(n)
return a
end
function luc{T<:Integer}(n::T, k::T)
NFib(n, k, luc_seeder)
end
Main
lo = 2
hi = 10
klim = 16
print("n-step Fibonacci for n = (", lo, ",", hi)
println(") up to k = ", klim, ":")
for i in 2:10
print(@sprintf("%5d => ", i))
for j in fib(i, klim)
print(j, " ")
end
println()
end
println()
print("n-step Rosetta Code Lucas for n = (", lo, ",", hi)
println(") up to k = ", klim, ":")
for i in 2:10
print(@sprintf("%5d => ", i))
for j in luc_rc(i, klim)
print(j, " ")
end
println()
end
println()
print("n-step MathWorld Lucas for n = (", lo, ",", hi)
println(") up to k = ", klim, ":")
for i in 2:10
print(@sprintf("%5d => ", i))
for j in luc(i, klim)
print(j, " ")
end
println()
end
- Output:
n-step Fibonacci for n = (2,10) up to k = 16:
2 => 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987
3 => 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768
4 => 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671
5 => 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624
6 => 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 15109
7 => 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 15808
8 => 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 16128
9 => 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 16272
10 => 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 16336
n-step Rosetta Code Lucas for n = (2,10) up to k = 16:
2 => 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364
3 => 2 1 3 6 10 19 35 64 118 217 399 734 1350 2483 4567 8400
4 => 2 1 3 6 12 22 43 83 160 308 594 1145 2207 4254 8200 15806
5 => 2 1 3 6 12 24 46 91 179 352 692 1360 2674 5257 10335 20318
6 => 2 1 3 6 12 24 48 94 187 371 736 1460 2896 5744 11394 22601
7 => 2 1 3 6 12 24 48 96 190 379 755 1504 2996 5968 11888 23680
8 => 2 1 3 6 12 24 48 96 192 382 763 1523 3040 6068 12112 24176
9 => 2 1 3 6 12 24 48 96 192 384 766 1531 3059 6112 12212 24400
10 => 2 1 3 6 12 24 48 96 192 384 768 1534 3067 6131 12256 24500
n-step MathWorld Lucas for n = (2,10) up to k = 16:
2 => 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207
3 => 1 3 7 11 21 39 71 131 241 443 815 1499 2757 5071 9327 17155
4 => 1 3 7 15 26 51 99 191 367 708 1365 2631 5071 9775 18842 36319
5 => 1 3 7 15 31 57 113 223 439 863 1695 3333 6553 12883 25327 49791
6 => 1 3 7 15 31 63 120 239 475 943 1871 3711 7359 14598 28957 57439
7 => 1 3 7 15 31 63 127 247 493 983 1959 3903 7775 15487 30847 61447
8 => 1 3 7 15 31 63 127 255 502 1003 2003 3999 7983 15935 31807 63487
9 => 1 3 7 15 31 63 127 255 511 1013 2025 4047 8087 16159 32287 64511
10 => 1 3 7 15 31 63 127 255 511 1023 2036 4071 8139 16271 32527 65023
Kotlin
// version 1.1.2
fun fibN(initial: IntArray, numTerms: Int) : IntArray {
val n = initial.size
require(n >= 2 && numTerms >= 0)
val fibs = initial.copyOf(numTerms)
if (numTerms <= n) return fibs
for (i in n until numTerms) {
var sum = 0
for (j in i - n until i) sum += fibs[j]
fibs[i] = sum
}
return fibs
}
fun main(args: Array<String>) {
val names = arrayOf("fibonacci", "tribonacci", "tetranacci", "pentanacci", "hexanacci",
"heptanacci", "octonacci", "nonanacci", "decanacci")
val initial = intArrayOf(1, 1, 2, 4, 8, 16, 32, 64, 128, 256)
println(" n name values")
var values = fibN(intArrayOf(2, 1), 15).joinToString(", ")
println("%2d %-10s %s".format(2, "lucas", values))
for (i in 0..8) {
values = fibN(initial.sliceArray(0 until i + 2), 15).joinToString(", ")
println("%2d %-10s %s".format(i + 2, names[i], values))
}
}
- Output:
n name values 2 lucas 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 2 fibonacci 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 3 tribonacci 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136 4 tetranacci 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536 5 pentanacci 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930 6 hexanacci 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617 7 heptanacci 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936 8 octonacci 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080 9 nonanacci 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144 10 decanacci 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172
Lua
function nStepFibs (seq, limit)
local iMax, sum = #seq - 1
while #seq < limit do
sum = 0
for i = 0, iMax do sum = sum + seq[#seq - i] end
table.insert(seq, sum)
end
return seq
end
local fibSeqs = {
{name = "Fibonacci", values = {1, 1} },
{name = "Tribonacci", values = {1, 1, 2} },
{name = "Tetranacci", values = {1, 1, 2, 4}},
{name = "Lucas", values = {2, 1} }
}
for _, sequence in pairs(fibSeqs) do
io.write(sequence.name .. ": ")
print(table.concat(nStepFibs(sequence.values, 10), " "))
end
- Output:
Fibonacci: 1 1 2 3 5 8 13 21 34 55 Tribonacci: 1 1 2 4 7 13 24 44 81 149 Tetranacci: 1 1 2 4 8 15 29 56 108 208 Lucas: 2 1 3 4 7 11 18 29 47 76
Maple
numSequence := proc(initValues :: Array)
local n, i, values;
n := numelems(initValues);
values := copy(initValues);
for i from (n+1) to 15 do
values(i) := add(values[i-n..i-1]);
end do;
return values;
end proc:
initValues := Array([1]):
for i from 2 to 10 do
initValues(i) := add(initValues):
printf ("nacci(%d): %a\n", i, convert(numSequence(initValues), list));
end do:
printf ("lucas: %a\n", convert(numSequence(Array([2, 1])), list));- Output:
nacci(2): [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610] nacci(3): [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136] nacci(4): [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536] nacci(5): [1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930] nacci(6): [1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617] nacci(7): [1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936] nacci(8): [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080] nacci(9): [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144] nacci(10): [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172] lucas: [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843]
Mathematica / Wolfram Language
f2=Function[{l,k},
Module[{n=Length@l,m},
m=SparseArray[{{i_,j_}/;i==1||i==j+1->1},{n,n}];
NestList[m.#&,l,k]]];
Table[Last/@f2[{1,1}~Join~Table[0,{n-2}],15+n][[-18;;]],{n,2,10}]//TableForm
Table[Last/@f2[{1,2}~Join~Table[0,{n-2}],15+n][[-18;;]],{n,2,10}]//TableForm
Output:
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624 26784 52656 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 15109 29970 59448 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 15808 31489 62725 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 16128 32192 64256 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 16272 32512 64960 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 16336 32656 65280 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571 2 1 3 6 10 19 35 64 118 217 399 734 1350 2483 4567 8400 15450 28417 2 1 3 6 12 22 43 83 160 308 594 1145 2207 4254 8200 15806 30467 58727 2 1 3 6 12 24 46 91 179 352 692 1360 2674 5257 10335 20318 39944 78528 2 1 3 6 12 24 48 94 187 371 736 1460 2896 5744 11394 22601 44831 88926 2 1 3 6 12 24 48 96 190 379 755 1504 2996 5968 11888 23680 47170 93961 2 1 3 6 12 24 48 96 192 382 763 1523 3040 6068 12112 24176 48256 96320 2 1 3 6 12 24 48 96 192 384 766 1531 3059 6112 12212 24400 48752 97408 2 1 3 6 12 24 48 96 192 384 768 1534 3067 6131 12256 24500 48976 97904
Nim
import sequtils, strutils
proc fiblike(start: seq[int]): auto =
var memo = start
proc fibber(n: int): int =
if n < memo.len:
return memo[n]
else:
var ans = 0
for i in n-start.len ..< n:
ans += fibber(i)
memo.add ans
return ans
return fibber
let fibo = fiblike(@[1,1])
echo toSeq(0..9).map(fibo)
let lucas = fiblike(@[2,1])
echo toSeq(0..9).map(lucas)
for n, name in items({2: "fibo", 3: "tribo", 4: "tetra", 5: "penta", 6: "hexa",
7: "hepta", 8: "octo", 9: "nona", 10: "deca"}):
var se = @[1]
for i in 0..n-2:
se.add(1 shl i)
let fibber = fiblike(se)
echo "n = ", align($n, 2), ", ", align(name, 5), "nacci -> ", toSeq(0..14).mapIt($fibber(it)).join(" "), " ..."
Output:
@[1, 1, 2, 3, 5, 8, 13, 21, 34, 55] @[2, 1, 3, 4, 7, 11, 18, 29, 47, 76] n = 2, fibonacci -> 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ... n = 3, tribonacci -> 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ... n = 4, tetranacci -> 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ... n = 5, pentanacci -> 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ... n = 6, hexanacci -> 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ... n = 7, heptanacci -> 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ... n = 8, octonacci -> 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ... n = 9, nonanacci -> 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ... n = 10, decanacci -> 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
Ol
We will use lazy lists, so can get any amount of n-nacci numbers.
(define (n-fib-iterator ll)
(cons (car ll)
(lambda ()
(n-fib-iterator (append (cdr ll) (list (fold + 0 ll)))))))
Testing:
(print "2, fibonacci : " (ltake (n-fib-iterator '(1 1)) 15))
(print "3, tribonacci: " (ltake (n-fib-iterator '(1 1 2)) 15))
(print "4, tetranacci: " (ltake (n-fib-iterator '(1 1 2 4)) 15))
(print "5, pentanacci: " (ltake (n-fib-iterator '(1 1 2 4 8)) 15))
(print "2, lucas : " (ltake (n-fib-iterator '(2 1)) 15))
; ==>
2, fibonacci : (1 1 2 3 5 8 13 21 34 55 89 144 233 377 610)
3, tribonacci: (1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136)
4, tetranacci: (1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536)
5, pentanacci: (1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930)
2, lucas : (2 1 3 4 7 11 18 29 47 76 123 199 322 521 843)
PARI/GP
The function gen generates code to generate a given number of terms of the k-th sequence. Of course there are other approaches.
Use genV if you prefer to supply a different starting vector.
gen(n)=k->my(v=vector(k,i,1));for(i=3,min(k,n),v[i]=2^(i-2));for(i=n+1,k,v[i]=sum(j=i-n,i-1,v[j]));v
genV(n)=v->for(i=3,min(#v,n),v[i]=2^(i-2));for(i=n+1,#v,v[i]=sum(j=i-n,i-1,v[j]));v
for(n=2,10,print(n"\t"gen(n)(10)))- Output:
2 [1, 1, 2, 3, 5, 8, 13, 21, 34, 55] 3 [1, 1, 2, 4, 7, 13, 24, 44, 81, 149] 4 [1, 1, 2, 4, 8, 15, 29, 56, 108, 208] 5 [1, 1, 2, 4, 8, 16, 31, 61, 120, 236] 6 [1, 1, 2, 4, 8, 16, 32, 63, 125, 248] 7 [1, 1, 2, 4, 8, 16, 32, 64, 127, 253] 8 [1, 1, 2, 4, 8, 16, 32, 64, 128, 255] 9 [1, 1, 2, 4, 8, 16, 32, 64, 128, 256] 10 [1, 1, 2, 4, 8, 16, 32, 64, 128, 256]
Pascal
program FibbonacciN (output);
type
TintArray = array of integer;
const
Name: array[2..11] of string = ('Fibonacci: ',
'Tribonacci: ',
'Tetranacci: ',
'Pentanacci: ',
'Hexanacci: ',
'Heptanacci: ',
'Octonacci: ',
'Nonanacci: ',
'Decanacci: ',
'Lucas: '
);
var
sequence: TintArray;
j, k: integer;
function CreateFibbo(n: integer): TintArray;
var
i: integer;
begin
setlength(CreateFibbo, n);
CreateFibbo[0] := 1;
CreateFibbo[1] := 1;
i := 2;
while i < n do
begin
CreateFibbo[i] := CreateFibbo[i-1] * 2;
inc(i);
end;
end;
procedure Fibbonacci(var start: TintArray);
const
No_of_examples = 11;
var
n, i, j: integer;
begin
n := length(start);
setlength(start, No_of_examples);
for i := n to high(start) do
begin
start[i] := 0;
for j := 1 to n do
start[i] := start[i] + start[i-j]
end;
end;
begin
for j := 2 to 10 do
begin
sequence := CreateFibbo(j);
Fibbonacci(sequence);
write (Name[j]);
for k := low(sequence) to high(sequence) do
write(sequence[k], ' ');
writeln;
end;
setlength(sequence, 2);
sequence[0] := 2;
sequence[1] := 1;
Fibbonacci(sequence);
write (Name[11]);
for k := low(sequence) to high(sequence) do
write(sequence[k], ' ');
writeln;
end.
Output:
% ./Fibbonacci Fibonacci: 1 1 2 3 5 8 13 21 34 55 89 Tribonacci: 1 1 2 4 7 13 24 44 81 149 274 Tetranacci: 1 1 2 4 8 15 29 56 108 208 401 Pentanacci: 1 1 2 4 8 16 31 61 120 236 464 Hexanacci: 1 1 2 4 8 16 32 63 125 248 492 Heptanacci: 1 1 2 4 8 16 32 64 127 253 504 Octonacci: 1 1 2 4 8 16 32 64 128 255 509 Nonanacci: 1 1 2 4 8 16 32 64 128 256 511 Decanacci: 1 1 2 4 8 16 32 64 128 256 512 Lucas: 2 1 3 4 7 11 18 29 47 76 123
Alternative
With the same output like above. A little bit like C++ alternative, but using only one idx and the observation,
that Sum[n] = 2*Sum[n-1]- Sum[n-stepSize].
There is no need to do so in Terms of speed, since fib(100) is out of reach using Uint64. Fib(n)/Fib(n-1) tends to the golden ratio = 1.618... 1.618^100 > 2^64
program FibbonacciN (output);
{$IFNDEF FPC}
{$APPTYPE CONSOLE}
{$ENDIF}
const
MAX_Nacci = 10;
No_of_examples = 11;// max 90; (golden ratio)^No < 2^64
Name: array[2..11] of string = ('Fibonacci: ',
'Tribonacci: ',
'Tetranacci: ',
'Pentanacci: ',
'Hexanacci: ',
'Heptanacci: ',
'Octonacci: ',
'Nonanacci: ',
'Decanacci: ',
'Lucas: '
);
type
tfibIdx = 0..MAX_Nacci;
tNacVal = Uint64;// longWord
tNacci = record
ncSum : tNacVal;
ncLastFib : array[tFibIdx] of tNacVal;
ncNextIdx : array[tFibIdx] of tFibIdx;
ncIdx : tFibIdx;
ncValue : tFibIdx;
end;
function CreateNacci(n: tFibIdx): TNacci;
var
i : tFibIdx;
sum :tNacVal;
begin
//With result do
with CreateNacci do
begin
ncLastFib[0] := 1;
ncLastFib[1] := 1;
For i := 2 to n-1 do
ncLastFib[i] := ncLastFib[i-1] * 2;
Sum := 0;
For i := 0 to n-1 do
sum := sum +ncLastFib[i];
ncSum := Sum;
//No need to do a compare
//inc(idx);
//if idx>= n then
// idx := 0;
//idx := nextIdx[idx]
For i := 0 to n-2 do
ncNextIdx[i] := i+1;
ncNextIdx[n-1] := 0;
ncIdx := 0;
end;
end;
function LehmerCreate:TNacci;
begin
with LehmerCreate do
begin
ncLastFib[0] := 2;
ncLastFib[1] := 1;
ncSum := 3;
ncNextIdx[0] := 1;
ncNextIdx[1] := 0;
ncIdx := 0;
end;
end;
function NextNacci(var Nacci:tNacci):tNacVal;
var
NewSum :tNacVal;
begin
with Nacci do
begin
NewSum := 2*ncSum- ncLastFib[ncIdx];
ncLastFib[ncIdx] := ncSum;
ncIdx := ncNextIdx[ncIdx];
NextNacci := ncSum;
ncSum := NewSum;
end;
end;
var
Nacci : tNacci;
j, k: integer;
BEGIN
for j := 2 to 10 do
begin
Nacci := CreateNacci(j);
write (Name[j]);
For k := 0 to j-1 do
write(Nacci.ncLastFib[k],' ');
For k := j to No_of_examples-1 do
write(NextNacci(Nacci),' ');
writeln;
end;
write (Name[11]);
j := 2;
Nacci := LehmerCreate;
For k := 0 to j-1 do
write(Nacci.ncLastFib[k],' ');
For k := j to No_of_examples-1 do
write(NextNacci(Nacci),' ');
writeln;
END.
PascalABC.NET
Unfold
I first define a high order function to generate infinite sequences given a lambda and a seed.
// unfold infinite sequences. Nigel Galloway: September 8th., 2022
function unfold<gN,gG>(n:Func<gG,(gN,gG)>; g:gG): sequence of gN;
begin
var (x,r):=n(g);
yield x;
yield sequence unfold(n,r);
end;
function unfold<gN,gG>(n:Func<array of gG,(gN,array of gG)>;params g:array of gG): sequence of gN := unfold(n,g);
The Task
Like the Pascal above but not iffy, not loopy, and not as long!
// Fibonacci n-step number sequences. Nigel Galloway: September 8th., 2022
var nFib:=function(n:array of biginteger): (biginteger,array of biginteger)->(n.First,n[1:].Append(n.Sum).ToArray);
begin
var fib:=unfold(nFib,1bi,1bi);
fib.Take(20).Println;
var tri:=unfold(nFib,fib.Take(3));
tri.Take(20).Println;
var tet:=unfold(nFib,tri.Take(4));
tet.Take(20).Println;
var pen:=unfold(nFib,tet.Take(5));
pen.Take(20).Println;
var hex:=unfold(nFib,pen.Take(6));
hex.Take(20).Println;
var hep:=unfold(nFib,hex.Take(7));
hep.Take(20).Println;
var oct:=unfold(nFib,hep.Take(8));
oct.Take(20).Println;
var non:=unfold(nFib,oct.Take(9));
non.Take(20).Println;
var dec:=unfold(nFib,non.Take(10));
dec.Take(20).Println;
var luc:=unfold(nFib,2bi,1bi);
luc.Take(20).Println;
end.
- Output:
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 35890 66012 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 76424 147312 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624 26784 52656 103519 203513 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 15109 29970 59448 117920 233904 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 15808 31489 62725 124946 248888 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 16128 32192 64256 128257 256005 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 16272 32512 64960 129792 259328 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 16336 32656 65280 130496 260864 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571 5778 9349
Perl
use strict;
use warnings;
use feature <say signatures>;
no warnings 'experimental';
use List::Util <max sum>;
sub fib_n ($n = 2, $xs = [1], $max = 100) {
my @xs = @$xs;
while ( $max > (my $len = @xs) ) {
push @xs, sum @xs[ max($len - $n, 0) .. $len-1 ];
}
@xs
}
say $_-1 . ': ' . join ' ', (fib_n $_)[0..19] for 2..10;
say "\nLucas: " . join ' ', fib_n(2, [2,1], 20);
- Output:
1: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 2: 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 35890 66012 3: 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 76424 147312 4: 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624 26784 52656 103519 203513 5: 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 15109 29970 59448 117920 233904 6: 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 15808 31489 62725 124946 248888 7: 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 16128 32192 64256 128257 256005 8: 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 16272 32512 64960 129792 259328 9: 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 16336 32656 65280 130496 260864 Lucas: 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571 5778 9349
Phix
with javascript_semantics function nacci_noo(integer n, s, l) if n<2 then return n+n*l end if if n=2 then return 1 end if atom res = nacci_noo(n-1,s,l) for i=2 to min(s,n-1) do res += nacci_noo(n-i,s,l) end for return res end function constant names = split("lucas fibo tribo tetra penta hexa hepta octo nona deca") sequence f = repeat(0,10) for i=1 to 4 do for j=1 to 10 do f[j] = nacci_noo(j,i+(i=1),i=1) end for printf(1,"%snacci: %v\n",{names[i],f}) end for
- Output:
lucasnacci: {2,1,3,4,7,11,18,29,47,76}
fibonacci: {1,1,2,3,5,8,13,21,34,55}
tribonacci: {1,1,2,4,7,13,24,44,81,149}
tetranacci: {1,1,2,4,8,15,29,56,108,208}
PHP
<?php
/**
* @author Elad Yosifon
*/
/**
* @param int $x
* @param array $series
* @param int $n
* @return array
*/
function fib_n_step($x, &$series = array(1, 1), $n = 15)
{
$count = count($series);
if($count > $x && $count == $n) // exit point
{
return $series;
}
if($count < $n)
{
if($count >= $x) // 4 or less
{
fib($series, $x, $count);
return fib_n_step($x, $series, $n);
}
else // 5 or more
{
while(count($series) < $x )
{
$count = count($series);
fib($series, $count, $count);
}
return fib_n_step($x, $series, $n);
}
}
return $series;
}
/**
* @param array $series
* @param int $n
* @param int $i
*/
function fib(&$series, $n, $i)
{
$end = 0;
for($j = $n; $j > 0; $j--)
{
$end += $series[$i-$j];
}
$series[$i] = $end;
}
/*=================== OUTPUT ============================*/
header('Content-Type: text/plain');
$steps = array(
'LUCAS' => array(2, array(2, 1)),
'FIBONACCI' => array(2, array(1, 1)),
'TRIBONACCI' => array(3, array(1, 1, 2)),
'TETRANACCI' => array(4, array(1, 1, 2, 4)),
'PENTANACCI' => array(5, array(1, 1, 2, 4)),
'HEXANACCI' => array(6, array(1, 1, 2, 4)),
'HEPTANACCI' => array(7, array(1, 1, 2, 4)),
'OCTONACCI' => array(8, array(1, 1, 2, 4)),
'NONANACCI' => array(9, array(1, 1, 2, 4)),
'DECANACCI' => array(10, array(1, 1, 2, 4)),
);
foreach($steps as $name=>$pair)
{
$ser = fib_n_step($pair[0],$pair[1]);
$n = count($ser)-1;
echo $name." => ".implode(',', $ser) . "\n";
}
- Output:
LUCAS => 2,1,3,4,7,11,18,29,47,76,123,199,322,521,843 FIBONACCI => 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610 TRIBONACCI => 1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136 TETRANACCI => 1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536 PENTANACCI => 1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930 HEXANACCI => 1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617 HEPTANACCI => 1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936 OCTONACCI => 1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080 NONANACCI => 1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144 DECANACCI => 1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172
PicoLisp
(de nacci (Init Cnt)
(let N (length Init)
(make
(made Init)
(do (- Cnt N)
(link (apply + (tail N (made)))) ) ) ) )Test:
# Fibonacci
: (nacci (1 1) 10)
-> (1 1 2 3 5 8 13 21 34 55)
# Tribonacci
: (nacci (1 1 2) 10)
-> (1 1 2 4 7 13 24 44 81 149)
# Tetranacci
: (nacci (1 1 2 4) 10)
-> (1 1 2 4 8 15 29 56 108 208)
# Lucas
: (nacci (2 1) 10)
-> (2 1 3 4 7 11 18 29 47 76)
# Decanacci
: (nacci (1 1 2 4 8 16 32 64 128 256) 15)
-> (1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172)PL/I
(subscriptrange, fixedoverflow, size):
n_step_Fibonacci: procedure options (main);
declare line character (100) varying;
declare (i, j, k) fixed binary;
put ('n-step Fibonacci series: Please type the initial values on one line:');
get edit (line) (L);
line = trim(line);
k = tally(line, ' ') - tally(line, ' ') + 1; /* count values */
begin;
declare (n(k), s) fixed decimal (15);
get string (line || ' ') list ( n );
if n(1) = 2 then put ('We have a Lusas series');
else put ('We have a ' || trim(k) || '-step Fibonacci series.');
put skip edit ( (trim(n(i)) do i = 1 to k) ) (a, x(1));
do j = k+1 to 20; /* In toto, generate 20 values in the series. */
s = sum(n); /* the next value in the series */
put edit (trim(s)) (x(1), a);
do i = lbound(n,1)+1 to k; /* Discard the oldest value */
n(i-1) = n(i);
end;
n(k) = s; /* and insert the new value */
end;
end;
end n_step_Fibonacci;Output:
We have a Lucas series. 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571 5778 9349 We have a 2-step Fibonacci series. 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 We have a 3-step Fibonacci series. 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 35890 66012 We have a 4-step Fibonacci series. 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 76424 147312 We have a 5-step Fibonacci series. 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624 26784 52656 103519 203513
Powershell
#Create generator of extended fibonaci
Function Get-ExtendedFibonaciGenerator($InitialValues ){
$Values = $InitialValues
{
#exhaust initial values first before calculating next values by summation
if ($InitialValues.Length -gt 0) {
$NextValue = $InitialValues[0]
$Script:InitialValues = $InitialValues | Select -Skip 1
return $NextValue
}
$NextValue = $Values | Measure-Object -Sum | Select -ExpandProperty Sum
$Script:Values = @($Values | Select-Object -Skip 1) + @($NextValue)
$NextValue
}.GetNewClosure()
}
Example of invocation to generate up to decanaci
$Name = 'fibo tribo tetra penta hexa hepta octo nona deca'.Split()
0..($Name.Length-1) | foreach { $Index = $_
$InitialValues = @(1) + @(foreach ($I In 0..$Index) { [Math]::Pow(2,$I) })
$Generator = Get-ExtendedFibonaciGenerator $InitialValues
[PSCustomObject] @{
n = $InitialValues.Length;
Name = "$($Name[$Index])naci";
Sequence = 1..15 | foreach { & $Generator } | Join-String -Separator ','
}
} | Format-Table -AutoSize
Sample output
n Name Sequence - ---- -------- 2 fibonaci 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610 3 tribonaci 1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136 4 tetranaci 1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536 5 pentanaci 1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930 6 hexanaci 1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617 7 heptanaci 1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936 8 octonaci 1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080 9 nonanaci 1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144 10 decanaci 1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172
PureBasic
Procedure.i FibonacciLike(k,n=2,p.s="",d.s=".")
Protected i,r
if k<0:ProcedureReturn 0:endif
if p.s
n=CountString(p.s,d.s)+1
for i=0 to n-1
if k=i:ProcedureReturn val(StringField(p.s,i+1,d.s)):endif
next
else
if k=0:ProcedureReturn 1:endif
if k=1:ProcedureReturn 1:endif
endif
for i=1 to n
r+FibonacciLike(k-i,n,p.s,d.s)
next
ProcedureReturn r
EndProcedure
; The fact that PureBasic supports default values for procedure parameters
; is very useful in a case such as this.
; Since:
; k=4
; Debug FibonacciLike(k) ;good old Fibonacci
; Debug FibonacciLike(k,3) ;here we specified n=3 [Tribonacci]
; Debug FibonacciLike(k,3,"1.1.2") ;using the default delimiter "."
; Debug FibonacciLike(k,3,"1,1,2",",") ;using a different delimiter ","
; the last three all produce the same result.
; as do the following two for the Lucas series:
; Debug FibonacciLike(k,2,"2.1") ;using the default delimiter "."
; Debug FibonacciLike(k,2,"2,1",",") ;using a different delimiter ","
m=10
t.s=lset("n",5)
for k=0 to m
t.s+lset(str(k),5)
next
Debug t.s
for n=2 to 10
t.s=lset(str(n),5)
for k=0 to m
t.s+lset(str(FibonacciLike(k,n)),5)
next
Debug t.s
next
Debug ""
p.s="2.1"
t.s=lset(p.s,5)
for k=0 to m
t.s+lset(str(FibonacciLike(k,n,p.s)),5)
next
Debug t.s
Debug ""
Sample Output
n 0 1 2 3 4 5 6 7 8 9 10 2 1 1 2 3 5 8 13 21 34 55 89 3 1 1 2 4 7 13 24 44 81 149 274 4 1 1 2 4 8 15 29 56 108 208 401 5 1 1 2 4 8 16 31 61 120 236 464 6 1 1 2 4 8 16 32 63 125 248 492 7 1 1 2 4 8 16 32 64 127 253 504 8 1 1 2 4 8 16 32 64 128 255 509 9 1 1 2 4 8 16 32 64 128 256 511 10 1 1 2 4 8 16 32 64 128 256 512 2.1 2 1 3 4 7 11 18 29 47 76 123
Python
Python: function returning a function
>>> def fiblike(start):
addnum = len(start)
memo = start[:]
def fibber(n):
try:
return memo[n]
except IndexError:
ans = sum(fibber(i) for i in range(n-addnum, n))
memo.append(ans)
return ans
return fibber
>>> fibo = fiblike([1,1])
>>> [fibo(i) for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> lucas = fiblike([2,1])
>>> [lucas(i) for i in range(10)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76]
>>> for n, name in zip(range(2,11), 'fibo tribo tetra penta hexa hepta octo nona deca'.split()) :
fibber = fiblike([1] + [2**i for i in range(n-1)])
print('n=%2i, %5snacci -> %s ...' % (n, name, ' '.join(str(fibber(i)) for i in range(15))))
n= 2, fibonacci -> 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...
n= 3, tribonacci -> 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ...
n= 4, tetranacci -> 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ...
n= 5, pentanacci -> 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ...
n= 6, hexanacci -> 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ...
n= 7, heptanacci -> 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ...
n= 8, octonacci -> 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ...
n= 9, nonanacci -> 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ...
n=10, decanacci -> 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
>>>
Python: Callable class
>>> class Fiblike():
def __init__(self, start):
self.addnum = len(start)
self.memo = start[:]
def __call__(self, n):
try:
return self.memo[n]
except IndexError:
ans = sum(self(i) for i in range(n-self.addnum, n))
self.memo.append(ans)
return ans
>>> fibo = Fiblike([1,1])
>>> [fibo(i) for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> lucas = Fiblike([2,1])
>>> [lucas(i) for i in range(10)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76]
>>> for n, name in zip(range(2,11), 'fibo tribo tetra penta hexa hepta octo nona deca'.split()) :
fibber = Fiblike([1] + [2**i for i in range(n-1)])
print('n=%2i, %5snacci -> %s ...' % (n, name, ' '.join(str(fibber(i)) for i in range(15))))
n= 2, fibonacci -> 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...
n= 3, tribonacci -> 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ...
n= 4, tetranacci -> 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ...
n= 5, pentanacci -> 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ...
n= 6, hexanacci -> 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ...
n= 7, heptanacci -> 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ...
n= 8, octonacci -> 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ...
n= 9, nonanacci -> 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ...
n=10, decanacci -> 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
>>>
Python: Generator
from itertools import islice, cycle
def fiblike(tail):
for x in tail:
yield x
for i in cycle(xrange(len(tail))):
tail[i] = x = sum(tail)
yield x
fibo = fiblike([1, 1])
print list(islice(fibo, 10))
lucas = fiblike([2, 1])
print list(islice(lucas, 10))
suffixes = "fibo tribo tetra penta hexa hepta octo nona deca"
for n, name in zip(xrange(2, 11), suffixes.split()):
fib = fiblike([1] + [2 ** i for i in xrange(n - 1)])
items = list(islice(fib, 15))
print "n=%2i, %5snacci -> %s ..." % (n, name, items)
- Output:
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55] [2, 1, 3, 4, 7, 11, 18, 29, 47, 76] n= 2, fibonacci -> [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610] ... n= 3, tribonacci -> [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136] ... n= 4, tetranacci -> [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536] ... n= 5, pentanacci -> [1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930] ... n= 6, hexanacci -> [1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617] ... n= 7, heptanacci -> [1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936] ... n= 8, octonacci -> [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080] ... n= 9, nonanacci -> [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144] ... n=10, decanacci -> [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172] ...
Python: Defined in terms of a generic anamorphism
Defining the Lucas series and the N-Step Fibonacci series in terms of unfoldr (dual to functools.reduce).
'''Fibonacci n-step number sequences'''
from itertools import chain, count, islice
# A000032 :: () -> [Int]
def A000032():
'''Non finite sequence of Lucas numbers.
'''
return unfoldr(recurrence(2))([2, 1])
# nStepFibonacci :: Int -> [Int]
def nStepFibonacci(n):
'''Non-finite series of N-step Fibonacci numbers,
defined by a recurrence relation.
'''
return unfoldr(recurrence(n))(
take(n)(
chain(
[1],
(2 ** i for i in count(0))
)
)
)
# recurrence :: Int -> [Int] -> Int
def recurrence(n):
'''Recurrence relation in Fibonacci and related series.
'''
def go(xs):
h, *t = xs
return h, t + [sum(take(n)(xs))]
return go
# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''First 15 terms each n-step Fibonacci(n) series
where n is drawn from [2..8]
'''
labels = "fibo tribo tetra penta hexa hepta octo nona deca"
table = list(
chain(
[['lucas:'] + [
str(x) for x in take(15)(A000032())]
],
map(
lambda k, n: list(
chain(
[k + 'nacci:'],
(
str(x) for x
in take(15)(nStepFibonacci(n))
)
)
),
labels.split(),
count(2)
)
)
)
print('Recurrence relation series:\n')
print(
spacedTable(table)
)
# ----------------------- GENERIC ------------------------
# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.
'''
def go(xs):
return (
xs[0:n]
if isinstance(xs, (list, tuple))
else list(islice(xs, n))
)
return go
# unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
def unfoldr(f):
'''Generic anamorphism.
A lazy (generator) list unfolded from a seed value by
repeated application of f until no residue remains.
Dual to fold/reduce.
f returns either None, or just (value, residue).
For a strict output value, wrap in list().
'''
def go(x):
valueResidue = f(x)
while None is not valueResidue:
yield valueResidue[0]
valueResidue = f(valueResidue[1])
return go
# ---------------------- FORMATTING ----------------------
# spacedTable :: [[String]] -> String
def spacedTable(rows):
columnWidths = [
max([len(x) for x in col])
for col in zip(*rows)
]
return '\n'.join([
' '.join(
map(
lambda x, w: x.rjust(w, ' '),
row, columnWidths
)
)
for row in rows
])
# MAIN ---
if __name__ == '__main__':
main()
- Output:
Recurrence relation series:
lucas: 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843
fibonacci: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
tribonacci: 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136
tetranacci: 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536
pentanacci: 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930
hexanacci: 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617
heptanacci: 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936
octonacci: 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080
nonanacci: 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144
decanacci: 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172
Quackery
[ 0 swap witheach + ] is sum ( [ --> n )
[ tuck size -
dup 0 < iff
[ split drop ]
else
[ dip [ dup size negate swap ]
times
[ over split
dup sum join join ]
nip ] ] is n-step ( n [ --> [ )
[ ' [ 1 1 ] n-step ] is fibonacci ( n --> [ )
[ ' [ 1 1 2 ] n-step ] is tribonacci ( n --> [ )
[ ' [ 1 1 2 4 ] n-step ] is tetranacci ( n --> [ )
[ ' [ 2 1 ] n-step ] is lucas ( n --> [ )
' [ fibonacci tribonacci tetranacci lucas ]
witheach
[ dup echo say ": " 10 swap do echo cr ]- Output:
fibonacci: [ 1 1 2 3 5 8 13 21 34 55 ] tribonacci: [ 1 1 2 4 7 13 24 44 81 149 ] tetranacci: [ 1 1 2 4 8 15 29 56 108 208 ] lucas: [ 2 1 3 4 7 11 18 29 47 76 ]
Racket
#lang racket
;; fib-list : [Listof Nat] x Nat -> [Listof Nat]
;; Given a non-empty list of natural numbers, the length of the list
;; becomes the size of the step; return the first n numbers of the
;; sequence; assume n >= (length lon)
(define (fib-list lon n)
(define len (length lon))
(reverse (for/fold ([lon (reverse lon)]) ([_ (in-range (- n len))])
(cons (apply + (take lon len)) lon))))
;; Show the series ...
(define (show-fibs name l)
(printf "~a: " name)
(for ([n (in-list (fib-list l 20))]) (printf "~a, " n))
(printf "...\n"))
;; ... with initial 2-powers lists
(for ([n (in-range 2 11)])
(show-fibs (format "~anacci" (case n [(2) 'fibo] [(3) 'tribo] [(4) 'tetra]
[(5) 'penta] [(6) 'hexa] [(7) 'hepta]
[(8) 'octo] [(9) 'nona] [(10) 'deca]))
(cons 1 (build-list (sub1 n) (curry expt 2)))))
;; and with an initial (2 1)
(show-fibs "lucas" '(2 1))
- Output:
fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, ... tribonacci: 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, ... tetranacci: 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, ... pentanacci: 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, ... hexanacci: 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, ... heptanacci: 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, ... octonacci: 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, ... nonanacci: 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, ... decanacci: 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280, 130496, 260864, ... lucas: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ...
Raku
(formerly Perl 6)
Lazy List with Closure
sub nacci ( $s = 2, :@start = (1,) ) {
my @seq = |@start, { state $n = +@start; @seq[ ($n - $s .. $n++ - 1).grep: * >= 0 ].sum } … *;
}
put "{.fmt: '%2d'}-nacci: ", nacci($_)[^20] for 2..12 ;
put "Lucas: ", nacci(:start(2,1))[^20];
- Output:
2-nacci: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 3-nacci: 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 35890 66012 4-nacci: 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 76424 147312 5-nacci: 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624 26784 52656 103519 203513 6-nacci: 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 15109 29970 59448 117920 233904 7-nacci: 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 15808 31489 62725 124946 248888 8-nacci: 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 16128 32192 64256 128257 256005 9-nacci: 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 16272 32512 64960 129792 259328 10-nacci: 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 16336 32656 65280 130496 260864 11-nacci: 1 1 2 4 8 16 32 64 128 256 512 1024 2047 4093 8184 16364 32720 65424 130816 261568 12-nacci: 1 1 2 4 8 16 32 64 128 256 512 1024 2048 4095 8189 16376 32748 65488 130960 261888 Lucas: 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571 5778 9349
Generative
A slightly more straight forward way of constructing a lazy list.
sub fib ($n, @xs is copy = [1]) {
flat gather {
take @xs[*];
loop {
take my $x = [+] @xs;
@xs.push: $x;
@xs.shift if @xs > $n;
}
}
}
for 2..10 -> $n {
say fib($n, [1])[^20];
}
say fib(2, [2,1])[^20];
REXX
/*REXX program calculates and displays a N-step Fibonacci sequence(s). */
parse arg FibName values /*allows a Fibonacci name, starter vals*/
if FibName\='' then do; call nStepFib FibName,values; signal done; end
/* [↓] no args specified, show a bunch*/
call nStepFib 'Lucas' , 2 1
call nStepFib 'fibonacci' , 1 1
call nStepFib 'tribonacci' , 1 1 2
call nStepFib 'tetranacci' , 1 1 2 4
call nStepFib 'pentanacci' , 1 1 2 4 8
call nStepFib 'hexanacci' , 1 1 2 4 8 16
call nStepFib 'heptanacci' , 1 1 2 4 8 16 32
call nStepFib 'octonacci' , 1 1 2 4 8 16 32 64
call nStepFib 'nonanacci' , 1 1 2 4 8 16 32 64 128
call nStepFib 'decanacci' , 1 1 2 4 8 16 32 64 128 256
call nStepFib 'undecanacci' , 1 1 2 4 8 16 32 64 128 256 512
call nStepFib 'dodecanacci' , 1 1 2 4 8 16 32 64 128 256 512 1024
call nStepFib '13th-order' , 1 1 2 4 8 16 32 64 128 256 512 1024 2048
done: exit /*stick a fork in it, we're all done. */
/*────────────────────────────────────────────────────────────────────────────*/
nStepFib: procedure; parse arg Fname,vals,m; if m=='' then m=30; L=
N=words(vals)
do pop=1 for N /*use N initial values. */
@.pop=word(vals,pop) /*populate initial numbers*/
end /*pop*/
do j=1 for m /*calculate M Fib numbers.*/
if j>N then do; @.j=0 /*initialize the sum to 0.*/
do k=j-N for N /*sum the last N numbers.*/
@.j=@.j+@.k /*add the [N-j]th number.*/
end /*k*/
end
L=L @.j /*append Fib number──►list*/
end /*j*/
say right(Fname,11)'[sum'right(N,3) "terms]:" strip(L) '···'
return
output when using the default input:
Lucas[sum 2 terms]: 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571 5778 9349 15127 24476 39603 64079 103682 167761 271443 439204 710647 1149851 ··· fibonacci[sum 2 terms]: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 ··· tribonacci[sum 3 terms]: 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 35890 66012 121415 223317 410744 755476 1389537 2555757 4700770 8646064 15902591 29249425 ··· tetranacci[sum 4 terms]: 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 76424 147312 283953 547337 1055026 2033628 3919944 7555935 14564533 28074040 54114452 104308960 ··· pentanacci[sum 5 terms]: 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624 26784 52656 103519 203513 400096 786568 1546352 3040048 5976577 11749641 23099186 45411804 89277256 175514464 ··· hexanacci[sum 6 terms]: 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 15109 29970 59448 117920 233904 463968 920319 1825529 3621088 7182728 14247536 28261168 56058368 111196417 220567305 ··· heptanacci[sum 7 terms]: 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 15808 31489 62725 124946 248888 495776 987568 1967200 3918592 7805695 15548665 30972384 61695880 122895984 244804400 ··· octonacci[sum 8 terms]: 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 16128 32192 64256 128257 256005 510994 1019960 2035872 4063664 8111200 16190208 32316160 64504063 128752121 256993248 ··· nonanacci[sum 9 terms]: 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 16272 32512 64960 129792 259328 518145 1035269 2068498 4132920 8257696 16499120 32965728 65866496 131603200 262947072 ··· decanacci[sum 10 terms]: 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 16336 32656 65280 130496 260864 521472 1042432 2083841 4165637 8327186 16646200 33276064 66519472 132973664 265816832 ··· undecanacci[sum 11 terms]: 1 1 2 4 8 16 32 64 128 256 512 1024 2047 4093 8184 16364 32720 65424 130816 261568 523008 1045760 2091008 4180992 8359937 16715781 33423378 66830392 133628064 267190704 ··· dodecanacci[sum 12 terms]: 1 1 2 4 8 16 32 64 128 256 512 1024 2048 4095 8189 16376 32748 65488 130960 261888 523712 1047296 2094336 4188160 8375296 16748544 33492993 66977797 133939218 267845688 ··· 13th-order[sum 13 terms]: 1 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8191 16381 32760 65516 131024 262032 524032 1048000 2095872 4191488 8382464 16763904 33525760 67047424 134086657 268156933 ···
Ring
# Project : Fibonacci n-step number sequences
f = list(12)
see "Fibonacci:" + nl
f2 = [1,1]
for nr2 = 1 to 10
see "" + f2[1] + " "
fibn(f2)
next
showarray(f2)
see " ..." + nl + nl
see "Tribonacci:" + nl
f3 = [1,1,2]
for nr3 = 1 to 9
see "" + f3[1] + " "
fibn(f3)
next
showarray(f3)
see " ..." + nl + nl
see "Tetranacci:" + nl
f4 = [1,1,2,4]
for nr4 = 1 to 8
see "" + f4[1] + " "
fibn(f4)
next
showarray(f4)
see " ..." + nl + nl
see "Lucas:" + nl
f5 = [2,1]
for nr5 = 1 to 10
see "" + f5[1] + " "
fibn(f5)
next
showarray(f5)
see " ..." + nl + nl
func fibn(fs)
s = sum(fs)
for i = 2 to len(fs)
fs[i-1] = fs[i]
next
fs[i-1] = s
return fs
func sum(arr)
sm = 0
for sn = 1 to len(arr)
sm = sm + arr[sn]
next
return sm
func showarray(fn)
svect = ""
for p = 1 to len(fn)
svect = svect + fn[p] + " "
next
see svectOutput:
Fibonacci: 1 1 2 3 5 8 13 21 34 55 89 144 ... Tribonacci: 1 1 2 4 7 13 24 44 81 149 274 504 ... Tetranacci: 1 1 2 4 8 15 29 56 108 208 401 773 ... Lucas: 2 1 3 4 7 11 18 29 47 76 123 199 ...
RPL
≪ OVER SIZE → len n
≪ LIST→
1 + len FOR j
n DUPN
2 n START + NEXT
NEXT len →LIST
≫ ≫ ‘NFIB’ STO
{1 1} 15 NFIB DUP 1 3 SUB 15 NFIB DUP 1 4 SUB 15 NFIB {2 1} 15 NFIB
- Output:
4: { 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 }
3: { 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 }
2: { 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 }
1: { 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 }
Ruby
def anynacci(start_sequence, count)
n = start_sequence.length # Get the n-step for the type of fibonacci sequence
result = start_sequence.dup # Create a new result array with the values copied from the array that was passed by reference
(count-n).times do # Loop for the remaining results up to count
result << result.last(n).sum # Get the last n element from result and append its total to Array
end
result
end
naccis = { lucas: [2,1],
fibonacci: [1,1],
tribonacci: [1,1,2],
tetranacci: [1,1,2,4],
pentanacci: [1,1,2,4,8],
hexanacci: [1,1,2,4,8,16],
heptanacci: [1,1,2,4,8,16,32],
octonacci: [1,1,2,4,8,16,32,64],
nonanacci: [1,1,2,4,8,16,32,64,128],
decanacci: [1,1,2,4,8,16,32,64,128,256] }
naccis.each {|name, seq| puts "%12s : %p" % [name, anynacci(seq, 15)]}
- Output:
lucas : [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843]
fibonacci : [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610]
tribonacci : [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136]
tetranacci : [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536]
pentanacci : [1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930]
hexanacci : [1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617]
heptanacci : [1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936]
octonacci : [1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080]
nonanacci : [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144]
decanacci : [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172]
Run BASIC
a = fib(" fibonacci ", "1,1")
a = fib("tribonacci ", "1,1,2")
a = fib("tetranacci ", "1,1,2,4")
a = fib(" pentanacc ", "1,1,2,4,8")
a = fib(" hexanacci ", "1,1,2,4,8,16")
a = fib(" lucas ", "2,1")
function fib(f$, s$)
dim f(20)
while word$(s$,b+1,",") <> ""
b = b + 1
f(b) = val(word$(s$,b,","))
wend
PRINT f$; "=>";
for i = b to 13 + b
print " "; f(i-b+1); ",";
for j = (i - b) + 1 to i
f(i+1) = f(i+1) + f(j)
next j
next i
print
end function- Output:
fibonacci => 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,
tribonacci => 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705,
tetranacci => 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872,
pentanacc => 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525,
hexanacci => 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840,
lucas => 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521,
Rust
struct GenFibonacci {
buf: Vec<u64>,
sum: u64,
idx: usize,
}
impl Iterator for GenFibonacci {
type Item = u64;
fn next(&mut self) -> Option<u64> {
let result = Some(self.sum);
self.sum -= self.buf[self.idx];
self.buf[self.idx] += self.sum;
self.sum += self.buf[self.idx];
self.idx = (self.idx + 1) % self.buf.len();
result
}
}
fn print(buf: Vec<u64>, len: usize) {
let mut sum = 0;
for &elt in buf.iter() { sum += elt; print!("\t{}", elt); }
let iter = GenFibonacci { buf: buf, sum: sum, idx: 0 };
for x in iter.take(len) {
print!("\t{}", x);
}
}
fn main() {
print!("Fib2:");
print(vec![1,1], 10 - 2);
print!("\nFib3:");
print(vec![1,1,2], 10 - 3);
print!("\nFib4:");
print(vec![1,1,2,4], 10 - 4);
print!("\nLucas:");
print(vec![2,1], 10 - 2);
}
Fib2: 1 1 2 3 5 8 13 21 34 55
Fib3: 1 1 2 4 7 13 24 44 81 149
Fib4: 1 1 2 4 8 15 29 56 108 208
Lucas: 2 1 3 4 7 11 18 29 47 76
Scala
Simple Solution
//we rely on implicit conversion from Int to BigInt.
//BigInt is preferable since the numbers get very big, very fast.
//(though for a small example of the first few numbers it's not needed)
def fibStream(init: BigInt*): LazyList[BigInt] = {
def inner(prev: Vector[BigInt]): LazyList[BigInt] = prev.head #:: inner(prev.tail :+ prev.sum)
inner(init.toVector)
}
Optimizing
//in the optimized version we don't compute values until it's needed.
//the unoptimized version, computed k elements ahead, where k being
//the number of elements to sum (fibonacci: k=2, tribonacci: k=3, ...).
def fib2Stream(init: BigInt*): LazyList[BigInt] = {
def inner(prev: Vector[BigInt]): LazyList[BigInt] = {
val sum = prev.sum
sum #:: inner(prev.tail :+ sum)
}
init.to(LazyList) #::: inner(init.toVector)
}
Optimizing Further
//instead of summing k elements each phase, we exploit the fact
//that the last element is already the sum of all k preceding elements
def fib3Stream(init: BigInt*): LazyList[BigInt] = {
def inner(prev: Vector[BigInt]): LazyList[BigInt] = {
val n = prev.last * 2 - prev.head
n #:: inner(prev.tail :+ n)
}
//last element must be the sum of k preceding elements, vector size should be k+1
val v = init.toVector :+ init.sum
v.to(LazyList) #::: inner(v)
}
Printing
println(s"Fibonacci: ${fibStream(1,1).take(10).mkString(",")}")
println(s"Tribonacci: ${fibStream(1,1,2).take(10).mkString(",")}")
println(s"Tetranacci: ${fibStream(1,1,2,4).take(10).mkString(",")}")
println(s"Lucas: ${fibStream(2,1).take(10).mkString(",")}")
- Output:
Fibonacci: 1,1,2,3,5,8,13,21,34,55 Tribonacci: 1,1,2,4,7,13,24,44,81,149 Tetranacci: 1,1,2,4,8,15,29,56,108,208 Lucas: 2,1,3,4,7,11,18,29,47,76
Note: In Scala, Stream is a lazy list. if you don't need the sequence saved in memory, just to iterate over members, you may convert the logic to use Iterator instead of Stream.
Scheme
(import (scheme base)
(scheme write)
(srfi 1))
;; uses n-step sequence formula to
;; continue lst until of length num
(define (n-fib lst num)
(let ((n (length lst)))
(do ((result (reverse lst)
(cons (fold + 0 (take result n))
result)))
((= num (length result)) (reverse result)))))
;; display examples
(do ((i 2 (+ 1 i)))
((> i 4) )
(display (string-append "n = "
(number->string i)
": "))
(display (n-fib (cons 1 (list-tabulate (- i 1) (lambda (n) (expt 2 n))))
15))
(newline))
(display "Lucas: ")
(display (n-fib '(2 1) 15))
(newline)
- Output:
n = 2: (1 1 2 3 5 8 13 21 34 55 89 144 233 377 610) n = 3: (1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136) n = 4: (1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536) Lucas: (2 1 3 4 7 11 18 29 47 76 123 199 322 521 843)
Seed7
$ include "seed7_05.s7i";
const func array integer: bonacci (in array integer: start, in integer: arity, in integer: length) is func
result
var array integer: bonacciSequence is 0 times 0;
local
var integer: sum is 0;
var integer: index is 0;
begin
bonacciSequence := start[.. length];
while length(bonacciSequence) < length do
sum := 0;
for index range max(1, length(bonacciSequence) - arity + 1) to length(bonacciSequence) do
sum +:= bonacciSequence[index];
end for;
bonacciSequence &:= [] (sum);
end while;
end func;
const proc: print (in string: name, in array integer: sequence) is func
local
var integer: index is 0;
begin
write((name <& ":") rpad 12);
for index range 1 to pred(length(sequence)) do
write(sequence[index] <& ", ");
end for;
writeln(sequence[length(sequence)]);
end func;
const proc: main is func
begin
print("Fibonacci", bonacci([] (1, 1), 2, 10));
print("Tribonacci", bonacci([] (1, 1), 3, 10));
print("Tetranacci", bonacci([] (1, 1), 4, 10));
print("Lucas", bonacci([] (2, 1), 2, 10));
end func;- Output:
Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 Tribonacci: 1, 1, 2, 4, 7, 13, 24, 44, 81, 149 Tetranacci: 1, 1, 2, 4, 8, 15, 29, 56, 108, 208 Lucas: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76
Sidef
func fib(n, xs=[1], k=20) {
loop {
var len = xs.len
len >= k && break
xs << xs.slice(max(0, len - n)).sum
}
return xs
}
for i in (2..10) {
say fib(i).join(' ')
}
say fib(2, [2, 1]).join(' ')
- Output:
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10609 19513 35890 66012 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 10671 20569 39648 76424 147312 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 13624 26784 52656 103519 203513 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 15109 29970 59448 117920 233904 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 15808 31489 62725 124946 248888 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 16128 32192 64256 128257 256005 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 16272 32512 64960 129792 259328 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 16336 32656 65280 130496 260864 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 1364 2207 3571 5778 9349
Using matrix exponentiation:
func fibonacci_matrix(k) is cached {
Matrix.build(k,k, {|i,j|
((i == k-1) || (i == j-1)) ? 1 : 0
})
}
func fibonacci_kth_order(n, k=2) {
var A = fibonacci_matrix(k)
(A**n)[0][-1]
}
for k in (2..9) {
say ("Fibonacci of k=#{k} order: ", (15+k).of { fibonacci_kth_order(_, k) })
}
- Output:
Fibonacci of k=2 order: [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987] Fibonacci of k=3 order: [0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768] Fibonacci of k=4 order: [0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671] Fibonacci of k=5 order: [0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624] Fibonacci of k=6 order: [0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109] Fibonacci of k=7 order: [0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808] Fibonacci of k=8 order: [0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128] Fibonacci of k=9 order: [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272]
Faster algorithm:
func fibonacci_kth_order (n, k = 2) {
return 0 if (n < k-1)
var f = (1..(k+1) -> map {|j|
j < k ? 2**j : 1
})
k += 1
for i in (2*(k-1) .. n) {
f[i%k] = (2*f[(i-1)%k] - f[i%k])
}
return f[n%k]
}
for k in (2..9) {
say ("Fibonacci of k=#{k} order: ", (15+k).of { fibonacci_kth_order(_, k) })
}
(same output as above)
Tailspin
templates fibonacciNstep&{N:}
templates next
@: $(1);
$(2..last)... -> @: $ + $@;
[ $(2..last)..., $@ ] !
end next
@: $;
1..$N -> #
<>
$@(1) !
@: $@ -> next;
end fibonacciNstep
[1,1] -> fibonacciNstep&{N:10} -> '$; ' -> !OUT::write
'
' -> !OUT::write
[1,1,2] -> fibonacciNstep&{N:10} -> '$; ' -> !OUT::write
'
' -> !OUT::write
[1,1,2,4] -> fibonacciNstep&{N:10} -> '$; ' -> !OUT::write
'
' -> !OUT::write
[2,1] -> fibonacciNstep&{N:10} -> '$; ' -> !OUT::write
'
' -> !OUT::write- Output:
1 1 2 3 5 8 13 21 34 55 1 1 2 4 7 13 24 44 81 149 1 1 2 4 8 15 29 56 108 208 2 1 3 4 7 11 18 29 47 76
Tcl
package require Tcl 8.6
proc fibber {args} {
coroutine fib[incr ::fibs]=[join $args ","] apply {fn {
set n [info coroutine]
foreach f $fn {
if {![yield $n]} return
set n $f
}
while {[yield $n]} {
set fn [linsert [lreplace $fn 0 0] end [set n [+ {*}$fn]]]
}
} ::tcl::mathop} $args
}
proc print10 cr {
for {set i 1} {$i <= 10} {incr i} {
lappend out [$cr true]
}
puts \[[join [lappend out ...] ", "]\]
$cr false
}
puts "FIBONACCI"
print10 [fibber 1 1]
puts "TRIBONACCI"
print10 [fibber 1 1 2]
puts "TETRANACCI"
print10 [fibber 1 1 2 4]
puts "LUCAS"
print10 [fibber 2 1]
- Output:
FIBONACCI [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...] TRIBONACCI [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ...] TETRANACCI [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ...] LUCAS [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...]
VBA
Option Explicit
Sub Main()
Dim temp$, T() As Long, i&
'Fibonacci:
T = Fibonacci_Step(1, 15, 1)
For i = LBound(T) To UBound(T)
temp = temp & ", " & T(i)
Next
Debug.Print "Fibonacci: " & Mid(temp, 3)
temp = ""
'Tribonacci:
T = Fibonacci_Step(1, 15, 2)
For i = LBound(T) To UBound(T)
temp = temp & ", " & T(i)
Next
Debug.Print "Tribonacci: " & Mid(temp, 3)
temp = ""
'Tetranacci:
T = Fibonacci_Step(1, 15, 3)
For i = LBound(T) To UBound(T)
temp = temp & ", " & T(i)
Next
Debug.Print "Tetranacci: " & Mid(temp, 3)
temp = ""
'Lucas:
T = Fibonacci_Step(1, 15, 1, 2)
For i = LBound(T) To UBound(T)
temp = temp & ", " & T(i)
Next
Debug.Print "Lucas: " & Mid(temp, 3)
temp = ""
End Sub
Private Function Fibonacci_Step(First As Long, Count As Long, S As Long, Optional Second As Long) As Long()
Dim T() As Long, R() As Long, i As Long, Su As Long, C As Long
If Second <> 0 Then S = 1
ReDim T(1 - S To Count)
For i = LBound(T) To 0
T(i) = 0
Next i
T(1) = IIf(Second <> 0, Second, 1)
T(2) = 1
For i = 3 To Count
Su = 0
C = S + 1
Do While C >= 0
Su = Su + T(i - C)
C = C - 1
Loop
T(i) = Su
Next
ReDim R(1 To Count)
For i = 1 To Count
R(i) = T(i)
Next
Fibonacci_Step = R
End Function
- Output:
Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 Tribonacci: 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136 Tetranacci: 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536 Lucas: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843
VBScript
'function arguments:
'init - initial series of the sequence(e.g. "1,1")
'rep - how many times the sequence repeats - init
Function generate_seq(init,rep)
token = Split(init,",")
step_count = UBound(token)
rep = rep - (UBound(token) + 1)
out = init
For i = 1 To rep
sum = 0
n = step_count
Do While n >= 0
sum = sum + token(UBound(token)-n)
n = n - 1
Loop
'add the next number to the sequence
ReDim Preserve token(UBound(token) + 1)
token(UBound(token)) = sum
out = out & "," & sum
Next
generate_seq = out
End Function
WScript.StdOut.Write "fibonacci: " & generate_seq("1,1",15)
WScript.StdOut.WriteLine
WScript.StdOut.Write "tribonacci: " & generate_seq("1,1,2",15)
WScript.StdOut.WriteLine
WScript.StdOut.Write "tetranacci: " & generate_seq("1,1,2,4",15)
WScript.StdOut.WriteLine
WScript.StdOut.Write "lucas: " & generate_seq("2,1",15)
WScript.StdOut.WriteLine
- Output:
fibonacci: 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610 tribonacci: 1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136 tetranacci: 1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536 lucas: 2,1,3,4,7,11,18,29,47,76,123,199,322,521,843
Visual Basic .NET
' Fibonacci n-step number sequences - VB.Net
Public Class FibonacciNstep
Const nmax = 20
Sub Main()
Dim bonacci As String() = {"", "", "Fibo", "tribo", "tetra", "penta", "hexa"}
Dim i As Integer
'Fibonacci:
For i = 2 To 6
Debug.Print(bonacci(i) & "nacci: " & FibonacciN(i, nmax))
Next i
'Lucas:
Debug.Print("Lucas: " & FibonacciN(2, nmax, 2))
End Sub 'Main
Private Function FibonacciN(iStep As Long, Count As Long, Optional First As Long = 0) As String
Dim i, j As Integer, Sigma As Long, c As String
Dim T(nmax) As Long
T(1) = IIf(First = 0, 1, First)
T(2) = 1
For i = 3 To Count
Sigma = 0
For j = i - 1 To i - iStep Step -1
If j > 0 Then
Sigma += T(j)
End If
Next j
T(i) = Sigma
Next i
c = ""
For i = 1 To nmax
c &= ", " & T(i)
Next i
Return Mid(c, 3)
End Function 'FibonacciN
End Class 'FibonacciNstep
- Output:
Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 tribonacci: 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012 tetranacci: 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312 pentanacci: 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513 hexanacci: 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904 Lucas: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349
V (Vlang)
fn fib_n(initial []int, num_terms int) []int {
n := initial.len
if n < 2 || num_terms < 0 {panic("Invalid argument(s).")}
if num_terms <= n {return initial}
mut fibs := []int{len:num_terms}
for i in 0..n {
fibs[i] = initial[i]
}
for i in n..num_terms {
mut sum := 0
for j in i-n..i {
sum = sum + fibs[j]
}
fibs[i] = sum
}
return fibs
}
fn main(){
names := [
"fibonacci", "tribonacci", "tetranacci", "pentanacci", "hexanacci",
"heptanacci", "octonacci", "nonanacci", "decanacci"
]
initial := [1, 1, 2, 4, 8, 16, 32, 64, 128, 256]
println(" n name values")
mut values := fib_n([2, 1], 15)
print(" 2 ${'lucas':-10}")
println(values.map('${it:4}').join(' '))
for i in 0..names.len {
values = fib_n(initial[0..i + 2], 15)
print("${i+2:2} ${names[i]:-10}")
println(values.map('${it:4}').join(' '))
}
}- Output:
n name values 2 lucas 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 2 fibonacci 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 3 tribonacci 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 4 tetranacci 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 5 pentanacci 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 6 hexanacci 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 7 heptanacci 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 8 octonacci 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 9 nonanacci 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 10 decanacci 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172
Wren
import "./fmt" for Fmt
var fibN = Fn.new { |initial, numTerms|
var n = initial.count
if (n < 2 || numTerms < 0) Fiber.abort("Invalid argument(s).")
if (numTerms <= n) return initial.toList
var fibs = List.filled(numTerms, 0)
for (i in 0...n) fibs[i] = initial[i]
for (i in n...numTerms) {
var sum = 0
for (j in i-n...i) sum = sum + fibs[j]
fibs[i] = sum
}
return fibs
}
var names = [
"fibonacci", "tribonacci", "tetranacci", "pentanacci", "hexanacci",
"heptanacci", "octonacci", "nonanacci", "decanacci"
]
var initial = [1, 1, 2, 4, 8, 16, 32, 64, 128, 256]
System.print(" n name values")
var values = fibN.call([2, 1], 15)
Fmt.write("$2d $-10s", 2, "lucas")
Fmt.aprint(values, 4, 0, "")
for (i in 0..8) {
values = fibN.call(initial[0...i + 2], 15)
Fmt.write("$2d $-10s", i + 2, names[i])
Fmt.aprint(values, 4, 0, "")
}
- Output:
n name values 2 lucas 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 2 fibonacci 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 3 tribonacci 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 4 tetranacci 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 5 pentanacci 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 6 hexanacci 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 7 heptanacci 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 8 octonacci 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 9 nonanacci 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 10 decanacci 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172
XPL0
include c:\cxpl\codes; \intrinsic 'code' declarations
proc Nacci(N, F0); \Generate Fibonacci N-step sequence
int N, \step size
F0; \array of first N values
int I, J;
def M = 10; \number of members in the sequence
int F(M); \Fibonacci sequence
[for I:= 0 to M-1 do \for all the members of the sequence...
[if I < N then F(I):= F0(I) \initialize sequence
else [F(I):= 0; \sum previous members to get member I
for J:= 1 to N do F(I):= F(I) + F(I-J);
];
IntOut(0, F(I)); ChOut(0, ^ );
];
CrLf(0);
];
[Text(0, " Fibonacci: "); Nacci(2, [1, 1]);
Text(0, "Tribonacci: "); Nacci(3, [1, 1, 2]);
Text(0, "Tetranacci: "); Nacci(4, [1, 1, 2, 4]);
Text(0, " Lucas: "); Nacci(2, [2, 1]);
]Output:
Fibonacci: 1 1 2 3 5 8 13 21 34 55
Tribonacci: 1 1 2 4 7 13 24 44 81 149
Tetranacci: 1 1 2 4 8 15 29 56 108 208
Lucas: 2 1 3 4 7 11 18 29 47 76
Yabasic
sub nStepFibs$(seq$, limit)
local iMax, sum, numb$(1), lim, i
lim = token(seq$, numb$(), ",")
redim numb$(limit)
seq$ = ""
iMax = lim - 1
while(lim < limit)
sum = 0
for i = 0 to iMax : sum = sum + val(numb$(lim - i)) : next
lim = lim + 1
numb$(lim) = str$(sum)
wend
for i = 0 to lim : seq$ = seq$ + " " + numb$(i) : next
return seq$
end sub
print "Fibonacci:", nStepFibs$("1,1", 10)
print "Tribonacci:", nStepFibs$("1,1,2", 10)
print "Tetranacci:", nStepFibs$("1,1,2,4", 10)
print "Lucas:", nStepFibs$("2,1", 10)zkl
fcn fibN(ns){ fcn(ns){ ns.append(ns.sum()).pop(0) }.fp(vm.arglist.copy()); }This stores the initial n terms of the sequence and returns a function that, at each call, appends the sum of the terms to the sequence then pops the leading value and returns it.
N:=15;
lucas:=fibN(2,1); do(N){ lucas().print(","); } println(); // Lucas
ns:=L(1); foreach _ in ([ns.len()+1..10]){ // Fibonacci n-step for 2 .. 10
ns.append(ns.sum()); // the inital values for the series
"%2d: ".fmt(ns.len()).print();
(N).pump(List,fibN(ns.xplode())).println();
}- Output:
2,1,3,4,7,11,18,29,47,76,123,199,322,521,843, 2: L(1,1,2,3,5,8,13,21,34,55,89,144,233,377,610) 3: L(1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136) 4: L(1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536) 5: L(1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930) 6: L(1,1,2,4,8,16,32,63,125,248,492,976,1936,3840,7617) 7: L(1,1,2,4,8,16,32,64,127,253,504,1004,2000,3984,7936) 8: L(1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080) 9: L(1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144) 10: L(1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088,8172)
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