# Fibonacci n-step number sequences

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:

1. For ${\displaystyle n=2}$ we have the Fibonacci sequence; with initial values ${\displaystyle [1,1]}$ and ${\displaystyle F_{k}^{2}=F_{k-1}^{2}+F_{k-2}^{2}}$
2. For ${\displaystyle n=3}$ we have the tribonacci sequence; with initial values ${\displaystyle [1,1,2]}$ and ${\displaystyle F_{k}^{3}=F_{k-1}^{3}+F_{k-2}^{3}+F_{k-3}^{3}}$
3. For ${\displaystyle n=4}$ we have the tetranacci sequence; with initial values ${\displaystyle [1,1,2,4]}$ and ${\displaystyle F_{k}^{4}=F_{k-1}^{4}+F_{k-2}^{4}+F_{k-3}^{4}+F_{k-4}^{4}}$
...
4. For general ${\displaystyle n>2}$ we have the Fibonacci ${\displaystyle n}$-step sequence - ${\displaystyle F_{k}^{n}}$; with initial values of the first ${\displaystyle n}$ values of the ${\displaystyle (n-1)}$'th Fibonacci ${\displaystyle n}$-step sequence ${\displaystyle F_{k}^{n-1}}$; and ${\displaystyle k}$'th value of this ${\displaystyle n}$'th sequence being ${\displaystyle F_{k}^{n}=\sum _{i=1}^{(n)}{F_{k-i}^{(n)}}}$

For small values of ${\displaystyle n}$, Greek numeric prefixes are sometimes used to individually name each series.

Fibonacci ${\displaystyle n}$-step sequences
${\displaystyle n}$ 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 ${\displaystyle n=2}$ but uses ${\displaystyle [2,1]}$ as its initial values.

1. Write a function to generate Fibonacci ${\displaystyle n}$-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.
2. Use this to print and show here at least the first ten members of the Fibo/tribo/tetra-nacci and Lucas sequences.

Also see

## 11l

Translation of: Python: Callable class
T Fiblike
[Int] memo

F (start)
.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
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: 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 Works with: Dyalog 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 Translation of: BASIC256 Works with: Chipmunk Basic version 3.6.4 100 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 Translation of: BASIC256 Works with: QBasic version 1.1 Works with: QuickBasic version 4.5 Works with: QB64 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 Translation of: BASIC256 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$_^ JH v1:v#\p03< _$.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

Translation of: PicoLisp
( ( 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
)
)
=   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));
}

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++)
{
}

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];
}

}
}

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
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;

}

int fibber(in size_t n) nothrow @safe {
if (n >= memo.length)
memo ~= iota(n - addNum, n).map!fibber.sum;
return memo[n];
}

10.iota.map!fibber.writeln;
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;

this(in T[] start) nothrow @safe {
this.memo = start.dup;
}

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, ...

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))))

(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

Translation of: Ruby
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 ).

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 ).

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 PROGRAM ## F# 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
write(6,'(a)',advance='no')'Enter the number of terms to sum: '
if ((n < 2) .or. (29 < n)) stop'Unreasonable!  Exit.'
write(6,'(a)',advance='no')'Show the the first how many terms of the sequence?  '
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)
write(6,*) 'Fine.  Enter the initial terms.'
do i=1, n
write(6, '(i2,a2)', advance = 'no') 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
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:

Translation of: Python
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 Works with: jq version 1.4 # Input: the initial array def nacci(arity; len): arity asarity | 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 ${\displaystyle n}$ values prior to ${\displaystyle k=1}$ rather than the first ${\displaystyle n}$ values. For example the (2 step) Fibonacci sequence is ${\displaystyle F_{k+1}=F_{k}+F_{k-1}}$ with ${\displaystyle F_{-1}=1}$ and ${\displaystyle F_{0}=0}$ rather than ${\displaystyle F_{1}=1}$ and ${\displaystyle F_{2}=1}$. 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 ${\displaystyle F_{1-n}=1}$ and ${\displaystyle F_{2-n}\ldots F_{0}=0}$. 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 ${\displaystyle F_{1-n}=3}$, ${\displaystyle F_{2-n}=-1}$ and, for ${\displaystyle n>2}$, ${\displaystyle F_{3-n}\ldots F_{0}=0}$.  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 ${\displaystyle F_{0}=n}$ and ${\displaystyle F_{1-n}\ldots F_{-1}=-1}$. 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 Translation of: Python 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 Works with: Free_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 Works with: Free_Pascal 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);


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
(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):
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.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).

Works with: Python version 3.7
'''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.

Works with: Rakudo version 2015.12
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.*/
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 svect

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  ...


## 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
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

Translation of: Perl
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

Works with: Tcl version 8.6
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

Translation of: Visual Basic
Works with: Visual Basic .NET version 2011
' 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)

Translation of: Wren
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 Translation of: Kotlin Library: Wren-fmt 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  $-10s", i + 2, names[i]) Fmt.aprint(values, 4, 0, "") }  Output:  n name values 2 lucas 2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 2 fibonacci 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 3 tribonacci 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 4 tetranacci 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 5 pentanacci 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 6 hexanacci 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 7 heptanacci 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 8 octonacci 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 9 nonanacci 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 10 decanacci 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172  ## XPL0 include c:\cxpl\codes; \intrinsic 'code' declarations proc Nacci(N, F0); \Generate Fibonacci N-step sequence int N, \step size F0; \array of first N values int I, J; def M = 10; \number of members in the sequence int F(M); \Fibonacci sequence [for I:= 0 to M-1 do \for all the members of the sequence... [if I < N then F(I):= F0(I) \initialize sequence else [F(I):= 0; \sum previous members to get member I for J:= 1 to N do F(I):= F(I) + F(I-J); ]; IntOut(0, F(I)); ChOut(0, ^ ); ]; CrLf(0); ]; [Text(0, " Fibonacci: "); Nacci(2, [1, 1]); Text(0, "Tribonacci: "); Nacci(3, [1, 1, 2]); Text(0, "Tetranacci: "); Nacci(4, [1, 1, 2, 4]); Text(0, " Lucas: "); Nacci(2, [2, 1]); ] Output:  Fibonacci: 1 1 2 3 5 8 13 21 34 55 Tribonacci: 1 1 2 4 7 13 24 44 81 149 Tetranacci: 1 1 2 4 8 15 29 56 108 208 Lucas: 2 1 3 4 7 11 18 29 47 76  ## Yabasic Translation of: Lua sub nStepFibs$(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)
`