Fusc sequence
The fusc integer sequence is defined as:
You are encouraged to solve this task according to the task description, using any language you may know.
- Definitions
- fusc(0) = 0
- fusc(1) = 1
- for n>1, the nth term is defined as:
- if n is even; fusc(n) = fusc(n/2)
- if n is odd; fusc(n) = fusc((n-1)/2) + fusc((n+1)/2)
Note that MathWorld's definition starts with unity, not zero. This task will be using the OEIS' version (above).
- An observation
-
- fusc(A) = fusc(B)
where A is some non-negative integer expressed in binary, and where B is the binary value of A reversed.
Fusc numbers are also known as:
- fusc function (named by Dijkstra, 1982)
- Stern's Diatomic series (although it starts with unity, not zero)
- Stern-Brocot sequence (although it starts with unity, not zero)
- Task
-
- show the first 61 fusc numbers (starting at zero) in a horizontal format.
- show the fusc number (and its index) whose length is greater than any previous fusc number length.
- (the length is the number of decimal digits when the fusc number is expressed in base ten.)
- show all numbers with commas (if appropriate).
- show all output here.
- Related task
- Also see
-
- the MathWorld entry: Stern's Diatomic Series.
- the OEIS entry: A2487.
11l
<lang 11l>F fusc(n)
V res = [0] * n res[1] = 1 L(i) 2 .< n res[i] = I i % 2 == 0 {res[i I/ 2]} E res[(i-1) I/ 2] + res[(i+1) I/ 2] R res
print(‘First 61 terms:’) print(fusc(61))
print() print(‘Points in the sequence where an item has more digits than any previous items:’) V f = fusc(20'000'000) V max_len = 0 L(i) 0 .< f.len
I String(f[i]).len > max_len max_len = String(f[i]).len print((i, f[i]))</lang>
- Output:
First 61 terms: [0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4] Points in the sequence where an item has more digits than any previous items: (0, 0) (37, 11) (1173, 108) (35499, 1076) (699051, 10946) (19573419, 103682)
Ada
<lang Ada>with Ada.Text_IO; with Ada.Integer_Text_IO;
procedure Show_Fusc is
generic Precalculate : Natural; package Fusc_Sequences is function Fusc (N : in Natural) return Natural; end Fusc_Sequences;
package body Fusc_Sequences is
Precalculated_Fusc : array (0 .. Precalculate) of Natural;
function Fusc_Slow (N : in Natural) return Natural is begin if N = 0 or N = 1 then return N; elsif N mod 2 = 0 then return Fusc_Slow (N / 2); else return Fusc_Slow ((N - 1) / 2) + Fusc_Slow ((N + 1) / 2); end if; end Fusc_Slow;
function Fusc (N : in Natural) return Natural is begin if N <= Precalculate then return Precalculated_Fusc (N); elsif N mod 2 = 0 then return Fusc (N / 2); else return Fusc ((N - 1) / 2) + Fusc ((N + 1) / 2); end if; end Fusc;
begin for N in Precalculated_Fusc'Range loop Precalculated_Fusc (N) := Fusc_Slow (N); end loop; end Fusc_Sequences;
package Fusc_Sequence is new Fusc_Sequences (Precalculate => 200_000);
function Fusc (N : in Natural) return Natural renames Fusc_Sequence.Fusc;
procedure Print_Small_Fuscs is use Ada.Text_IO; begin Put_Line ("First 61 numbers in the fusc sequence:"); for N in 0 .. 60 loop Put (Fusc (N)'Image); Put (" "); end loop; New_Line; end Print_Small_Fuscs;
procedure Print_Large_Fuscs (High : in Natural) is use Ada.Text_IO; use Ada.Integer_Text_IO; subtype N_Range is Natural range Natural'First .. High; F : Natural; Len : Natural; Max_Len : Natural := 0; Placeholder : String := " n fusc(n)"; Image_N : String renames Placeholder (1 .. 8); Image_Fusc : String renames Placeholder (10 .. Placeholder'Last); begin New_Line; Put_Line ("Printing all largest Fusc numbers upto " & High'Image); Put_Line (Placeholder);
for N in N_Range loop F := Fusc (N); Len := F'Image'Length;
if Len > Max_Len then Max_Len := Len; Put (Image_N, N); Put (Image_Fusc, F); Put (Placeholder); New_Line; end if; end loop; end Print_Large_Fuscs;
begin
Print_Small_Fuscs; Print_Large_Fuscs (High => 20_000_000);
end Show_Fusc;</lang>
- Output:
First 61 numbers in the fusc sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Printing all largest Fusc numbers upto 20000000 n fusc(n) 0 0 37 11 1173 108 35499 1076 699051 10946 19573419 103682
ALGOL 68
<lang algol68>BEGIN
# calculate some members of the fusc sequence # # f0 = 0, f1 = 1, fn = f(n/2) if n even # # = f(n-1)/2) + f((n+1)/2) if n odd #
# constructs an array of the first n elements of the fusc sequence # PROC fusc sequence = ( INT n )[]INT: BEGIN [ 0 : n ]INT a; IF n > 0 THEN a[ 0 ] := 0; IF n > 1 THEN a[ 1 ] := 1; INT i2 := 1; FOR i FROM 2 BY 2 TO n - 1 DO a[ i ] := a[ i2 ]; a[ i + 1 ] := a[ # j - i # i2 ] + a[ # ( j + 1 ) OVER 2 # i2 + 1 ]; i2 +:= 1 OD FI FI; a[ 0 : n - 1 AT 0 ] END ; # fusc #
[]INT f = fusc sequence( 800 000 ); FOR i FROM 0 TO 60 DO print( ( " ", whole( f[ i ], 0 ) ) ) OD; print( ( newline ) ); # find the lowest elements of the sequence that have 1, 2, 3, etc. digits # print( ( "Sequence elements where number of digits of the value increase:", newline ) ); print( ( " n fusc(n)", newline ) ); INT digit power := 0; FOR i FROM LWB f TO UPB f DO IF f[ i ] >= digit power THEN # found the first number with this many digits # print( ( whole( i, -8 ), " ", whole( f[ i ], -10 ), newline ) ); IF digit power = 0 THEN digit power := 1 FI; digit power *:= 10 FI OD
END</lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Sequence elements where number of digits of the value increase: n fusc(n) 0 0 37 11 1173 108 35499 1076 699051 10946
AppleScript
<lang applescript>on fusc(n)
if (n < 2) then return n else if (n mod 2 is 0) then return fusc(n div 2) else return fusc((n - 1) div 2) + fusc((n + 1) div 2) end if
end fusc
set sequence to {} set longestSoFar to 0 repeat with i from 0 to 60
set fuscNumber to fusc(i) set end of sequence to fuscNumber set len to (count (fuscNumber as text)) if (len > longestSoFar) then set longestSoFar to len set firstLongest to fuscNumber set indexThereof to i + 1 -- AppleScript indices are 1-based. end if
end repeat
return {sequence:sequence, firstLongest:firstLongest, indexThereof:indexThereof}</lang>
- Output:
{sequence:{0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4}, firstLongest:11, indexThereof:38}
Or defining generators, both for a non-finite stream of Fusc terms, and for the sequence of the first Fusc terms of each decimal magnitude:
<lang applescript>-- fusc :: [Int] on fusc()
-- Terms of the Fusc sequence -- OEIS A2487 script go on |λ|(step) set {isEven, n, xxs} to step set x to item 1 of xxs if isEven then set nxt to n + x {not isEven, nxt, xxs & {nxt}} else {not isEven, x, rest of xxs & {x}} end if end |λ| end script appendGen(gen({0, 1}), ¬ fmapGen(my snd, iterate(go, {true, 1, {1}})))
end fusc
TEST ---------------------------
on run
unlines({¬ "First 61 terms:", ¬ showList(take(61, fusc())), ¬ "", ¬ "First term of each decimal magnitude:", ¬ "(Index, Term):"} & ¬ map(showTuple, take(4, firstFuscOfEachMagnitude())))
end run
FIRST FUSC OF EACH DECIMAL MAGNITUDE -----------
-- firstFuscOfEachMagnitude :: [(Int, Int)] on firstFuscOfEachMagnitude()
-- [(Index, Term)] list of of the first Fusc -- terms of each decimal magnitude. script property e : -1 property i : 0 on |λ|() set e to 1 + e set p to 10 ^ e set v to fuscTerm(i) repeat until p ≤ v set i to 1 + i set v to fuscTerm(i) end repeat {i, v} end |λ| end script
end firstFuscOfEachMagnitude
-- fuscTerm :: Int -> Int
on fuscTerm(n)
-- Nth term (zero-indexed) of the Fusc series. script go on |λ|(i) if 0 = i then {1, 0} else set {x, y} to |λ|(i div 2) if 0 = i mod 2 then {x + y, y} else {x, x + y} end if end if end |λ| end script tell go if 1 > n then 0 else item 1 of |λ|(n - 1) end if end tell
end fuscTerm
GENERIC FUNCTIONS --------------------
-- appendGen (++) :: Gen [a] -> Gen [a] -> Gen [a] on appendGen(xs, ys)
script property vs : xs on |λ|() set v to |λ|() of vs if missing value is not v then v else set vs to ys |λ|() of ys end if end |λ| end script
end appendGen
-- fmapGen <$> :: (a -> b) -> Gen [a] -> Gen [b]
on fmapGen(f, gen)
script property g : mReturn(f) on |λ|() set v to gen's |λ|() if v is missing value then v else g's |λ|(v) end if end |λ| end script
end fmapGen
-- intercalate :: String -> [String] -> String
on intercalate(delim, xs)
set {dlm, my text item delimiters} to ¬ {my text item delimiters, delim} set s to xs as text set my text item delimiters to dlm s
end intercalate
-- iterate :: (a -> a) -> a -> Gen [a]
on iterate(f, x)
script property v : missing value property g : mReturn(f)'s |λ| on |λ|() if missing value is v then set v to x else set v to g(v) end if return v end |λ| end script
end iterate
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
-- The list obtained by applying f -- to each element of 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
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
-- 2nd class handler function lifted into 1st class script wrapper. if script is class of f then f else script property |λ| : f end script end if
end mReturn
-- gen :: [a] -> Gen a
on gen(xs)
script go property lng : length of xs property i : 0 on |λ|() if i ≥ lng then missing value else set i to 1 + i item i of xs end if end |λ| end script
end gen
-- showList :: [a] -> String
on showList(xs)
"[" & intercalate(", ", my map(my str, xs)) & "]"
end showList
-- showTuple :: (,) -> String
on showTuple(xs)
"(" & intercalate(", ", my map(my str, xs)) & ")"
end showTuple
-- snd :: (a, b) -> b
on snd(tpl)
if class of tpl is record then |2| of tpl else item 2 of tpl end if
end snd
-- str :: a -> String
on str(x)
x as string
end str
-- take :: Int -> [a] -> [a]
-- take :: Int -> String -> String
on take(n, xs)
set c to class of xs if list is c then if 0 < n then items 1 thru min(n, length of xs) of xs else {} end if else if string is c then if 0 < n then text 1 thru min(n, length of xs) of xs else "" end if else if script is c then set ys to {} repeat with i from 1 to n set v to |λ|() of xs if missing value is v then return ys else set end of ys to v end if end repeat return ys else missing value end if
end take
-- unlines :: [String] -> String
on unlines(xs)
-- A single string formed by the intercalation -- of a list of strings with the newline character. set {dlm, my text item delimiters} to ¬ {my text item delimiters, linefeed} set s to xs as text set my text item delimiters to dlm s
end unlines</lang>
- Output:
First 61 terms: [0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4] First term of each decimal magnitude: (Index, Term): (0, 0) (37, 11) (1173, 108) (35499, 1076)
Arturo
<lang rebol>fusc: function [n][
if? or? n=0 n=1 -> n else [ if? 0=n%2 -> fusc n/2 else -> (fusc (n-1)/2) + fusc (n+1)/2 ]
]
print "The first 61 Fusc numbers:" print map 0..61 => fusc
print "\nThe Fusc numbers whose lengths are greater than those of previous Fusc numbers:" print " n fusc(n)" print "--------- ---------" maxLength: 0
loop 0..40000 'i [
f: fusc i l: size to :string f if l > maxLength [ maxLength: l print [ pad to :string i 9 pad to :string f 9 ] ]
]</lang>
- Output:
The first 61 Fusc numbers: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 9 The Fusc numbers whose lengths are greater than those of previous Fusc numbers: n fusc(n) --------- --------- 0 0 37 11 1173 108 35499 1076
AWK
<lang AWK>
- syntax: GAWK -f FUSC_SEQUENCE.AWK
- converted from C
BEGIN {
for (i=0; i<61; i++) { printf("%d ",fusc(i)) } printf("\n") print("fusc numbers whose length is greater than any previous fusc number length") printf("%9s %9s\n","fusc","index") for (i=0; i<=700000; i++) { f = fusc(i) leng = num_leng(f) if (leng > max_leng) { max_leng = leng printf("%9s %9s\n",commatize(f),commatize(i)) } } exit(0)
} function commatize(x, num) {
if (x < 0) { return "-" commatize(-x) } x = int(x) num = sprintf("%d.",x) while (num ~ /^[0-9][0-9][0-9][0-9]/) { sub(/[0-9][0-9][0-9][,.]/,",&",num) } sub(/\.$/,"",num) return(num)
} function fusc(n) {
if (n == 0 || n == 1) { return(n) } else if (n % 2 == 0) { return fusc(n/2) } else { return fusc((n-1)/2) + fusc((n+1)/2) }
} function num_leng(n, sum) {
sum = 1 while (n > 9) { n = int(n/10) sum++ } return(sum)
} </lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 fusc numbers whose length is greater than any previous fusc number length fusc index 0 0 11 37 108 1,173 1,076 35,499 10,946 699,051
BQN
Works in: CBQN
Fusc
computes fusc numbers iteratively.
<lang bqn>Fusc ← {
{ 𝕩∾+´(⍷(⌈∾⌊)2÷˜≠𝕩)⊑¨<𝕩 }⍟(𝕩-2)↕2
}
•Show Fusc 61
•Show >⟨"Index"‿"Number"⟩∾{((1+↕4)⊐˜(⌊1+10⋆⁼1⌈|)¨𝕩){𝕨∾𝕨⊑𝕩}¨<𝕩} Fusc 99999</lang> <lang bqn>⟨ 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 ⟩ ┌─ ╵ "Index" "Number"
0 0 37 11 1173 108 35499 1076 ┘</lang>
C
<lang C>
- include<limits.h>
- include<stdio.h>
int fusc(int n){
if(n==0||n==1) return n; else if(n%2==0) return fusc(n/2); else return fusc((n-1)/2) + fusc((n+1)/2);
}
int numLen(int n){
int sum = 1;
while(n>9){ n = n/10; sum++; }
return sum;
}
void printLargeFuscs(int limit){
int i,f,len,maxLen = 1;
printf("\n\nPrinting all largest Fusc numbers upto %d \nIndex-------Value",limit);
for(i=0;i<=limit;i++){ f = fusc(i); len = numLen(f);
if(len>maxLen){ maxLen = len; printf("\n%5d%12d",i,f); } }
}
int main()
{
int i;
printf("Index-------Value"); for(i=0;i<61;i++) printf("\n%5d%12d",i,fusc(i)); printLargeFuscs(INT_MAX); return 0;
} </lang> Prints first 61 Fusc numbers followed by the largest numbers :
Index-------Value 0 0 1 1 2 1 3 2 4 1 5 3 6 2 7 3 8 1 9 4 10 3 11 5 12 2 13 5 14 3 15 4 16 1 17 5 18 4 19 7 20 3 21 8 22 5 23 7 24 2 25 7 26 5 27 8 28 3 29 7 30 4 31 5 32 1 33 6 34 5 35 9 36 4 37 11 38 7 39 10 40 3 41 11 42 8 43 13 44 5 45 12 46 7 47 9 48 2 49 9 50 7 51 12 52 5 53 13 54 8 55 11 56 3 57 10 58 7 59 11 60 4 Printing all largest Fusc numbers upto 2147483647 Index-------Value 37 11 1173 108 35499 1076 699051 10946 103682 19573419 1010747 615164587
C#
<lang csharp>using System; using System.Collections.Generic;
static class program {
static int n = 61; static List<int> l = new List<int>() { 0, 1 };
static int fusc(int n) { if (n < l.Count) return l[n]; int f = (n & 1) == 0 ? l[n >> 1] : l[(n - 1) >> 1] + l[(n + 1) >> 1]; l.Add(f); return f; }
static void Main(string[] args) { bool lst = true; int w = -1, c = 0, t; string fs = "{0,11:n0} {1,-9:n0}", res = ""; Console.WriteLine("First {0} numbers in the fusc sequence:", n); for (int i = 0; i < int.MaxValue; i++) { int f = fusc(i); if (lst) { if (i < 61) Console.Write("{0} ", f); else { lst = false; Console.WriteLine(); Console.WriteLine("Points in the sequence where an item has more digits than any previous items:"); Console.WriteLine(fs, "Index\\", "/Value"); Console.WriteLine(res); res = ""; } } if ((t = f.ToString().Length) > w) { w = t; res += (res == "" ? "" : "\n") + string.Format(fs, i, f); if (!lst) { Console.WriteLine(res); res = ""; } if (++c > 5) break; } } l.Clear(); }
}</lang>
- Output:
First 61 numbers in the fusc sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Points in the sequence where an item has more digits than any previous items: Index\ /Value 0 0 37 11 1,173 108 35,499 1,076 699,051 10,946 19,573,419 103,682
C++
<lang cpp>#include <iomanip>
- include <iostream>
- include <limits>
- include <sstream>
- include <vector>
const int n = 61; std::vector<int> l{ 0, 1 };
int fusc(int n) {
if (n < l.size()) return l[n]; int f = (n & 1) == 0 ? l[n >> 1] : l[(n - 1) >> 1] + l[(n + 1) >> 1]; l.push_back(f); return f;
}
int main() {
bool lst = true; int w = -1; int c = 0; int t; std::string res; std::cout << "First " << n << " numbers in the fusc sequence:\n"; for (int i = 0; i < INT32_MAX; i++) { int f = fusc(i); if (lst) { if (i < 61) { std::cout << f << ' '; } else { lst = false; std::cout << "\nPoints in the sequence where an item has more digits than any previous items:\n"; std::cout << std::setw(11) << "Index\\" << " " << std::left << std::setw(9) << "/Value\n"; std::cout << res << '\n'; res = ""; } } std::stringstream ss; ss << f; t = ss.str().length(); ss.str(""); ss.clear(); if (t > w) { w = t; res += (res == "" ? "" : "\n"); ss << std::setw(11) << i << " " << std::left << std::setw(9) << f; res += ss.str(); if (!lst) { std::cout << res << '\n'; res = ""; } if (++c > 5) { break; } } } return 0;
}</lang>
- Output:
First 61 numbers in the fusc sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Points in the sequence where an item has more digits than any previous items: Index\ /Value 0 0 37 11 1173 108 35499 1076 699051 10946 19573419 103682
D
Built-in Memoization
<lang d>import std.functional, std.stdio, std.format, std.conv;
ulong fusc(ulong n) =>
memoize!fuscImp(n);
ulong fuscImp(ulong n) =>
( n < 2 ) ? n : ( n % 2 == 0 ) ? memoize!fuscImp( n/2 ) : memoize!fuscImp( (n-1)/2 ) + memoize!fuscImp( (n+1)/2 );
void main() {
const N_FIRST=61; const MAX_N_DIGITS=5; format!"First %d fusc numbers: "(N_FIRST).write; foreach( n; 0..N_FIRST ) n.fusc.format!"%d ".write; writeln; format!"First fusc number for each number of digits (1 to %d digits):"(MAX_N_DIGITS).writeln; for(auto n=0, ndigits=0; ndigits<MAX_N_DIGITS; n++) if( n.fusc.to!string.length > ndigits ){ format!"fusc(%d)=%d"( n, n.fusc ).writeln; ndigits = n.fusc.to!string.length.to!int; }
}</lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 0: 0 37: 11 1173: 108 35499: 1076 699051: 10946
Manual memoization
<lang d>import std.stdio, std.format, std.conv;
int[] fusc_cache = [0, 1]; int fusc(int n) {
// Ensure cache contains all missing numbers until n for(auto i=fusc_cache.length;i<=n;i++) fusc_cache ~= i%2 == 0 ? fusc_cache[i/2] : fusc_cache[(i-1)/2] + fusc_cache[(i + 1)/2]; // Solve using cache return fusc_cache[n];
}
void main() {
const N_FIRST=61; const MAX_N_DIGITS=6; foreach( n; 0..N_FIRST ) n.fusc.format!"%d ".write; writeln; for(int n=0, ndigits=0; ndigits<MAX_N_DIGITS; n++) if( n.fusc.to!string.length > ndigits ){ format!"%d: %d"( n, n.fusc ).writeln; ndigits = n.fusc.to!string.length.to!int; }
}</lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 0: 0 37: 11 1173: 108 35499: 1076 699051: 10946 19573419: 103682
Delphi
See Pascal.
F#
The Function
<lang fsharp> // Generate the fusc sequence. Nigel Galloway: March 20th., 2019 let fG n=seq{for (n,g) in Seq.append n [1] |> Seq.pairwise do yield n; yield n+g} let fusc=seq{yield 0; yield! Seq.unfold(fun n->Some(n,fG n))(seq[1])|>Seq.concat}|> Seq.mapi(fun n g->(n,g)) </lang>
The Tasks
- Print first 62 elements
<lang fsharp> fusc |> Seq.take 61 |> Seq.iter(fun(_,g)->printf "%d " g); printfn "" </lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
- Show the fusc number (and its index) whose length is greater than any previous fusc number length
The first 6 take only 10 secs so let me be more ambitious <lang fsharp> let fN=let mutable n=0 in (fun (_,g)->if g>=n then n<-pown 10 (string g).Length; true else false) fusc |> Seq.filter fN |> Seq.take 7 |> Seq.iter(fun(n,g)->printfn "fusc %d -> %d" n g) </lang>
- Output:
fusc 0 -> 0 fusc 37 -> 11 fusc 1173 -> 108 fusc 35499 -> 1076 fusc 699051 -> 10946 fusc 19573419 -> 103682 fusc 615164587 -> 1010747 Real: 00:06:03.801, CPU: 00:06:03.140, GC gen0: 21336, gen1: 0
Factor
<lang factor>USING: arrays assocs formatting io kernel make math math.parser math.ranges namespaces prettyprint sequences tools.memory.private ; IN: rosetta-code.fusc
<PRIVATE
- (fusc) ( n -- seq )
[ 2 ] dip [a,b) [ 0 , 1 , [ [ building get ] dip dup even? [ 2/ swap nth ] [ [ 1 - 2/ ] [ 1 + 2/ ] 2bi [ swap nth ] 2bi@ + ] if , ] each ] { } make ;
- increases ( seq -- assoc )
[ 0 ] dip [ [ 2array 2dup first number>string length < [ [ 1 + ] [ , ] bi* ] [ drop ] if ] each-index ] { } make nip ;
PRIVATE>
- fusc ( n -- seq )
dup 3 < [ { 0 1 } swap head ] [ (fusc) ] if ;
- fusc-demo ( -- )
"First 61 fusc numbers:" print 61 fusc [ pprint bl ] each nl nl "Fusc numbers with more digits than all previous ones:" print "Value Index\n====== =======" print 1,000,000 fusc increases [ [ commas ] bi@ "%-6s %-7s\n" printf ] assoc-each ;
MAIN: fusc-demo</lang>
- Output:
First 61 fusc numbers: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Fusc numbers with more digits than all previous ones: Value Index ====== ======= 0 0 11 37 108 1,173 1,076 35,499 10,946 699,051
FreeBASIC
<lang freebasic>' version 01-03-2019 ' compile with: fbc -s console
- Define max 20000000
Dim Shared As UInteger f(max)
Sub fusc
f(0) = 0 f(1) = 1
For n As UInteger = 2 To max If n And 1 Then f(n) = f((n -1) \ 2) + f((n +1) \ 2) Else f(n) = f(n \ 2) End If Next
End Sub
' ------=< MAIN >=------
Dim As UInteger i, d Dim As String fs
fusc
For i = 0 To 60
Print f(i); " ";
Next
Print : Print Print " Index Value" For i = 0 To max
If f(i) >= d Then Print Using "###########," ; i; f(i) If d = 0 Then d = 1 d *= 10 End If
Next
' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Index Value 0 0 37 11 1,173 108 35,499 1,076 699,051 10,946 19,573,419 103,682
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.
In this page you can see the program(s) related to this task and their results.
Go
<lang go>package main
import (
"fmt" "strconv"
)
func fusc(n int) []int {
if n <= 0 { return []int{} } if n == 1 { return []int{0} } res := make([]int, n) res[0] = 0 res[1] = 1 for i := 2; i < n; i++ { if i%2 == 0 { res[i] = res[i/2] } else { res[i] = res[(i-1)/2] + res[(i+1)/2] } } return res
}
func fuscMaxLen(n int) [][2]int {
maxLen := -1 maxFusc := -1 f := fusc(n) var res [][2]int for i := 0; i < n; i++ { if f[i] <= maxFusc { continue // avoid expensive strconv operation where possible } maxFusc = f[i] le := len(strconv.Itoa(f[i])) if le > maxLen { res = append(res, [2]int{i, f[i]}) maxLen = le } } return res
}
func commatize(n int) string {
s := fmt.Sprintf("%d", n) if n < 0 { s = s[1:] } le := len(s) for i := le - 3; i >= 1; i -= 3 { s = s[0:i] + "," + s[i:] } if n >= 0 { return s } return "-" + s
}
func main() {
fmt.Println("The first 61 fusc numbers are:") fmt.Println(fusc(61)) fmt.Println("\nThe fusc numbers whose length > any previous fusc number length are:") res := fuscMaxLen(20000000) // examine first twenty million numbers say for i := 0; i < len(res); i++ { fmt.Printf("%7s (index %10s)\n", commatize(res[i][1]), commatize(res[i][0])) }
}</lang>
- Output:
The first 61 fusc numbers are: [0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4] The fusc numbers whose length > any previous fusc number length are: 0 (index 0) 11 (index 37) 108 (index 1,173) 1,076 (index 35,499) 10,946 (index 699,051) 103,682 (index 19,573,419)
Groovy
<lang groovy>class FuscSequence {
static void main(String[] args) { println("Show the first 61 fusc numbers (starting at zero) in a horizontal format") for (int n = 0; n < 61; n++) { printf("%,d ", fusc[n]) }
println() println() println("Show the fusc number (and its index) whose length is greater than any previous fusc number length.") int start = 0 for (int i = 0; i <= 5; i++) { int val = i != 0 ? (int) Math.pow(10, i) : -1 for (int j = start; j < FUSC_MAX; j++) { if (fusc[j] > val) { printf("fusc[%,d] = %,d%n", j, fusc[j]) start = j break } } } }
private static final int FUSC_MAX = 30000000 private static int[] fusc = new int[FUSC_MAX]
static { fusc[0] = 0 fusc[1] = 1 for (int n = 2; n < FUSC_MAX; n++) { int n2 = (int) (n / 2) int n2m = (int) ((n - 1) / 2) int n2p = (int) ((n + 1) / 2) fusc[n] = n % 2 == 0 ? fusc[n2] : fusc[n2m] + fusc[n2p] } }
}</lang>
- Output:
Show the first 61 fusc numbers (starting at zero) in a horizontal format 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Show the fusc number (and its index) whose length is greater than any previous fusc number length. fusc[0] = 0 fusc[37] = 11 fusc[1,173] = 108 fusc[35,499] = 1,076 fusc[699,051] = 10,946 fusc[19,573,419] = 103,682
Haskell
<lang haskell>---------------------- FUSC SEQUENCE ---------------------
fusc :: Int -> Int fusc i
| 1 > i = 0 | otherwise = fst $ go (pred i) where go n | 0 == n = (1, 0) | even n = (x + y, y) | otherwise = (x, x + y) where (x, y) = go (div n 2)
TEST -------------------------
main :: IO () main = do
putStrLn "First 61 terms:" print $ fusc <$> [0 .. 60] putStrLn "\n(Index, Value):" mapM_ print $ take 5 widths
widths :: [(Int, Int)] widths =
fmap (\(_, i, x) -> (i, x)) (iterate nxtWidth (2, 0, 0))
nxtWidth :: (Int, Int, Int) -> (Int, Int, Int) nxtWidth (w, i, v) = (succ w, j, x)
where fi = (,) <*> fusc (j, x) = until ((w <=) . length . show . snd) (fi . succ . fst) (fi i)</lang>
- Output:
First 61 terms: [0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4] (Index, Value): (0,0) (37,11) (1173,108) (35499,1076) (699051,10946)
Another version using infinite list: <lang haskell>zipWithLazy :: (a -> b -> c) -> [a] -> [b] -> [c] zipWithLazy f ~(x : xs) ~(y : ys) =
f x y : zipWithLazy f xs ys
fuscs :: [Integer] fuscs = 0 : s
where s = 1 : concat (zipWithLazy f s (tail s)) f x y = [x, x + y]
widths :: [(Int, Integer)] widths = map head $ scanl f (zip [0 ..] fuscs) [2 ..]
where f fis w = dropWhile ((< w) . length . show . snd) fis
main :: IO () main = do
putStrLn "First 61 terms:" print $ take 61 fuscs putStrLn "\n(Index, Value):" mapM_ print $ take 5 widths</lang>
- Output:
First 61 terms: [0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4] (Index, Value): (0,0) (37,11) (1173,108) (35499,1076) (699051,10946)
J
<lang J> fusc_term =: ({~ -:@#)`([: +/ ({~ ([: -: _1 1 + #)))@.(2 | #) fusc =: (, fusc_term)@:]^:[ 0 1"_
NB. show the first 61 fusc numbers (starting at zero) in a horizontal format. 61 {. fusc 70
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
9!:17]2 2 NB. specify bottom right position in box
FUSC =: fusc 99999 DIGITS =: ; ([: # 10&#.inv)&.> FUSC
(;: 'index value') ,. <"0(,: {&A) DIGITS i. 1 2 3 4
┌─────┬─┬──┬────┬─────┐ │index│0│37│1173│35499│ ├─────┼─┼──┼────┼─────┤ │value│0│11│ 108│ 1076│ └─────┴─┴──┴────┴─────┘
</lang>
Java
<lang Java>
public class FuscSequence {
public static void main(String[] args) { System.out.println("Show the first 61 fusc numbers (starting at zero) in a horizontal format"); for ( int n = 0 ; n < 61 ; n++ ) { System.out.printf("%,d ", fusc[n]); } System.out.printf("%n%nShow the fusc number (and its index) whose length is greater than any previous fusc number length.%n"); int start = 0; for (int i = 0 ; i <= 5 ; i++ ) { int val = i != 0 ? (int) Math.pow(10, i) : -1; for ( int j = start ; j < FUSC_MAX ; j++ ) { if ( fusc[j] > val ) { System.out.printf("fusc[%,d] = %,d%n", j, fusc[j] ); start = j; break; } } } } private static final int FUSC_MAX = 30000000; private static int[] fusc = new int[FUSC_MAX];
static { fusc[0] = 0; fusc[1] = 1; for ( int n = 2 ; n < FUSC_MAX ; n++ ) { fusc[n] = (n % 2 == 0 ? fusc[n/2] : fusc[(n-1)/2] + fusc[(n+1)/2]); } }
} </lang>
- Output:
Show the first 61 fusc numbers (starting at zero) in a horizontal format 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Show the fusc number (and its index) whose length is greater than any previous fusc number length. fusc[0] = 0 fusc[37] = 11 fusc[1,173] = 108 fusc[35,499] = 1,076 fusc[699,051] = 10,946 fusc[19,573,419] = 103,682
JavaScript
Functional
A composition of pure generic functions:
<lang javascript>(() => {
'use strict';
const main = () => {
// fusc :: Int -> Int const fusc = i => { const go = n => 0 === n ? ( [1, 0] ) : (() => { const [x, y] = go(quot(n, 2)); return even(n) ? ( [x + y, y] ) : [x, x + y]; })(); return 1 > i ? ( 0 ) : fst(go(i - 1)); };
// firstWidths :: Int -> [(Int, Int)] const firstWidths = n => { const nxtWidth = xs => { const fi = fanArrow(fusc, id), [w, i, v] = head(xs), [x, j] = Array.from(until( v => w <= fst(v).toString().length, v => fi(succ(snd(v))), fi(i) )); return cons( [succ(w), j, x], xs ); }; return until( x => n < fst(fst(x)), nxtWidth, 2, 0, 0 ); };
return unlines([ 'First 61 terms:', '[' + map(fusc, enumFromTo(0, 60)).join(',') + ']', , '(Index, Value):', unlines(map( ([i, x]) => '(' + i + ', ' + x + ')', foldl( (a, x) => cons(tail(x), a), [], firstWidths(5) ) )) ]); };
// GENERIC FUNCTIONS ----------------------------
// Tuple (,) :: a -> b -> (a, b) const Tuple = (a, b) => ({ type: 'Tuple', '0': a, '1': b, length: 2 });
// cons :: a -> [a] -> [a] const cons = (x, xs) => Array.isArray(xs) ? ( [x].concat(xs) ) : 'GeneratorFunction' !== xs.constructor.constructor.name ? ( x + xs ) : ( // Existing generator wrapped with one additional element function*() { yield x; let nxt = xs.next() while (!nxt.done) { yield nxt.value; nxt = xs.next(); } } )();
// enumFromTo :: Enum a => a -> a -> [a] const enumFromTo = (m, n) => { const [x, y] = [m, n].map(fromEnum), b = x + ('number' !== typeof m ? 0 : m - x); return Array.from({ length: 1 + (y - x) }, (_, i) => toEnum(m)(b + i)); };
// even :: Int -> Bool const even = n => 0 === n % 2;
// Compose a function from a simple value to a tuple of // the separate outputs of two different functions
// fanArrow (&&&) :: (a -> b) -> (a -> c) -> (a -> (b, c)) const fanArrow = (f, g) => x => Tuple(f(x), g(x));
// foldl :: (a -> b -> a) -> a -> [b] -> a const foldl = (f, a, xs) => xs.reduce(f, a);
// fromEnum :: Enum a => a -> Int const fromEnum = x => typeof x !== 'string' ? ( x.constructor === Object ? ( x.value ) : parseInt(Number(x)) ) : x.codePointAt(0);
// fst :: (a, b) -> a const fst = tpl => tpl[0];
// head :: [a] -> a const head = xs => xs.length ? xs[0] : undefined;
// id :: a -> a const id = x => x;
// map :: (a -> b) -> [a] -> [b] const map = (f, xs) => (Array.isArray(xs) ? ( xs ) : xs.split()).map(f);
// quot :: Int -> Int -> Int const quot = (n, m) => Math.floor(n / m);
// snd :: (a, b) -> b const snd = tpl => tpl[1];
// succ :: Enum a => a -> a const succ = x => { const t = typeof x; return 'number' !== t ? (() => { const [i, mx] = [x, maxBound(x)].map(fromEnum); return i < mx ? ( toEnum(x)(1 + i) ) : Error('succ :: enum out of range.') })() : x < Number.MAX_SAFE_INTEGER ? ( 1 + x ) : Error('succ :: Num out of range.') };
// tail :: [a] -> [a] const tail = xs => 0 < xs.length ? xs.slice(1) : [];
// The first argument is a sample of the type // allowing the function to make the right mapping
// toEnum :: a -> Int -> a const toEnum = e => x => { const m = e.enum, f = { 'number': Number, 'string': String.fromCodePoint, 'boolean': Boolean } [typeof e]; return f ? ( f(x) ) : m[m[x]]; };
// unlines :: [String] -> String const unlines = xs => xs.join('\n');
// until :: (a -> Bool) -> (a -> a) -> a -> a const until = (p, f, x) => { let v = x; while (!p(v)) v = f(v); return v; };
// MAIN --- return main();
})();</lang>
- Output:
First 61 terms: [0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4] (Index, Value): (0, 0) (37, 11) (1173, 108) (35499, 1076) (699051, 10946)
Julia
<lang julia>using Memoize, Formatting
@memoize function sternbrocot(n)
if n < 2 return n elseif iseven(n) return sternbrocot(div(n, 2)) else m = div(n - 1, 2) return sternbrocot(m) + sternbrocot(m + 1) end
end
function fusclengths(N=100000000)
println("sequence number : fusc value") maxlen = 0 for i in 0:N x = sternbrocot(i) if (len = length(string(x))) > maxlen println(lpad(format(i, commas=true), 15), " : ", format(x, commas=true)) maxlen = len end end
end
println("The first 61 fusc numbers are: ", [sternbrocot(x) for x in 0:60]) fusclengths()
</lang>
- Output:
The first 61 fusc numbers are: [0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4] sequence number : fusc value 0 : 0 37 : 11 1,173 : 108 35,499 : 1,076 699,051 : 10,946 19,573,419 : 103,682
Kotlin
<lang scala>// Version 1.3.21
fun fusc(n: Int): IntArray {
if (n <= 0) return intArrayOf() if (n == 1) return intArrayOf(0) val res = IntArray(n) res[1] = 1 for (i in 2 until n) { if (i % 2 == 0) { res[i] = res[i / 2] } else { res[i] = res[(i - 1) / 2] + res[(i + 1) / 2] } } return res
}
fun fuscMaxLen(n: Int): List<Pair<Int, Int>> {
var maxLen = -1 var maxFusc = -1 val f = fusc(n) val res = mutableListOf<Pair<Int, Int>>() for (i in 0 until n) { if (f[i] <= maxFusc) continue // avoid string conversion maxFusc = f[i] val len = f[i].toString().length if (len > maxLen) { res.add(Pair(i, f[i])) maxLen = len } } return res
}
fun main() {
println("The first 61 fusc numbers are:") println(fusc(61).asList()) println("\nThe fusc numbers whose length > any previous fusc number length are:") val res = fuscMaxLen(20_000_000) // examine first 20 million numbers say for (r in res) { System.out.printf("%,7d (index %,10d)\n", r.second, r.first) }
}</lang>
- Output:
The first 61 fusc numbers are: [0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4] The fusc numbers whose length > any previous fusc number length are: 0 (index 0) 11 (index 37) 108 (index 1,173) 1,076 (index 35,499) 10,946 (index 699,051) 103,682 (index 19,573,419)
Lua
<lang lua>function fusc(n)
n = math.floor(n) if n == 0 or n == 1 then return n elseif n % 2 == 0 then return fusc(n / 2) else return fusc((n - 1) / 2) + fusc((n + 1) / 2) end
end
function numLen(n)
local sum = 1 while n > 9 do n = math.floor(n / 10) sum = sum + 1 end return sum
end
function printLargeFuscs(limit)
print("Printing all largest Fusc numbers up to " .. limit) print("Index-------Value") local maxLen = 1 for i=0,limit do local f = fusc(i) local le = numLen(f) if le > maxLen then maxLen = le print(string.format("%5d%12d", i, f)) end end
end
function main()
print("Index-------Value") for i=0,60 do print(string.format("%5d%12d", i, fusc(i))) end printLargeFuscs(math.pow(2, 31) - 1)
end
main()</lang>
- Output:
Index-------Value 0 0 1 1 2 1 3 2 4 1 5 3 6 2 7 3 8 1 9 4 10 3 11 5 12 2 13 5 14 3 15 4 16 1 17 5 18 4 19 7 20 3 21 8 22 5 23 7 24 2 25 7 26 5 27 8 28 3 29 7 30 4 31 5 32 1 33 6 34 5 35 9 36 4 37 11 38 7 39 10 40 3 41 11 42 8 43 13 44 5 45 12 46 7 47 9 48 2 49 9 50 7 51 12 52 5 53 13 54 8 55 11 56 3 57 10 58 7 59 11 60 4 Printing all largest Fusc numbers up to 2147483647 Index-------Value 37 11 1173 108 35499 1076 699051 10946
Mathematica / Wolfram Language
<lang Mathematica>ClearAll[Fusc] Fusc[0] := 0 Fusc[1] := 1 Fusc[n_] := Fusc[n] = If[EvenQ[n], Fusc[n/2], Fusc[(n - 1)/2] + Fusc[(n + 1)/2]] Fusc /@ Range[0, 60] res = Template:0, 1; i = 0; PrintTemporary[Dynamic[{res, i}]]; While[Length[res] < 6,
f = Fusc[i]; If[IntegerLength[res-1, -1] < IntegerLength[f], AppendTo[res, {i, f}] ]; i++; ];
res</lang>
- Output:
{0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4} {{0, 1}, {37, 11}, {1173, 108}, {35499, 1076}, {699051, 10946}, {19573419, 103682}}
Nim
Using recursive procedure
This is the simplest way to compute the sequence, but not very efficient here as we compute several times the same values. The algorithm could be improved by using a cache to keep the values. <lang nim>import strformat
func fusc(n: int): int =
if n == 0 or n == 1: n elif n mod 2 == 0: fusc(n div 2) else: fusc((n - 1) div 2) + fusc((n + 1) div 2)
echo "The first 61 Fusc numbers:" for i in 0..61:
write(stdout, fmt"{fusc(i)} ")
echo "\n\nThe Fusc numbers whose lengths are greater than those of previous Fusc numbers:" echo fmt" n fusc(n)" echo "--------- ---------" var maxLength = 0 for i in 0..700_000:
var f = fusc(i) var length = len($f) if length > maxLength: maxLength = length echo fmt"{i:9} {f:9}"</lang>
- Output:
The first 61 Fusc numbers: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 9 The Fusc numbers whose lengths are greater than those of previous Fusc numbers: n fusc(n) --------- --------- 0 0 37 11 1173 108 35499 1076 699051 10946
Using iterators and double queues (deques)
This is a translation of the Python procedural algorithm, using iterators instead of generators. It allows to compute the seven first Fusc numbers whose lengths are greater than those of previous Fusc numbers. <lang Nim>import deques, strformat
iterator fusc(): int =
var q = [1].toDeque() yield 0 yield 1
while true: var val = q.popFirst() q.addLast(val) yield val
val += q[0] q.addLast(val) yield val
iterator longestFusc(): tuple[idx, val: int] =
var sofar = 0 var i = -1 for f in fusc(): inc i if f >= sofar: yield (i, f) sofar = if sofar == 0: 10 else: 10 * sofar
- ———————————————————————————————————————————————————————————————————————————————————————————————————
const
MaxFusc = 61 LongestCount = 7
echo &"First {MaxFusc}:" var i = -1 for f in fusc():
inc i stdout.write f if i == MaxFusc: echo "" break stdout.write ' '
echo "\nLength records:" var count = 0 for (i, f) in longestFusc():
inc count echo &"fusc({i}) = {f}" if count == LongestCount: break</lang>
- Output:
First 61: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 9 Length records: fusc(0) = 0 fusc(37) = 11 fusc(1173) = 108 fusc(35499) = 1076 fusc(699051) = 10946 fusc(19573419) = 103682 fusc(615164587) = 1010747
Pascal
Using dynamic array.To speed things up using Pointer. Found the indices of a specific base to oszillating.Tried power of phi with more success 11 ~ phi^5 <lang pascal>program fusc; uses
sysutils;
const {$IFDEF FPC}
MaxIdx = 1253 * 1000 * 1000; //19573420; // must be even
{$ELSE}
// Dynamics arrays in Delphi cann't be to large MaxIdx = 19573420; {$ENDIF}
type
tFuscElem = LongWord; tFusc = array of tFuscElem;
var
FuscField : tFusc;
function commatize(n:NativeUint):string; var
l,i : NativeUint;
begin
str(n,result); l := length(result); //no commatize if l < 4 then exit; //new length i := l+ (l-1) DIV 3; setlength(result,i); //copy chars to the right place While i <> l do Begin result[i]:= result[l];result[i-1]:= result[l-1]; result[i-2]:= result[l-2];result[i-3]:= ','; dec(i,4);dec(l,3); end;
end;
procedure OutFusc(StartIdx,EndIdx :NativeInt;const FF:tFusc); Begin
IF StartIdx < Low(FF) then StartIdx :=Low(FF); IF EndIdx > High(FF) then EndIdx := High(FF); For StartIdx := StartIdx to EndIdx do write(FF[StartIdx],' '); writeln;
end;
procedure FuscCalc(var FF:tFusc); var
pFFn,pFFi : ^tFuscElem; i,n,sum : NativeUint;
Begin
FF[0]:= 0; FF[1]:= 1; n := 2; i := 1; pFFn := @FF[n]; pFFi := @FF[i]; sum := pFFi^; while n <= MaxIdx-2 do begin //even pFFn^ := sum;//FF[n] := FF[i]; //odd inc(pFFi);//FF[i+1] inc(pFFn);//FF[n+1] sum := sum+pFFi^; pFFn^:= sum; //FF[n+1] := FF[i]+FF[i+1]; sum := pFFi^; inc(pFFn); inc(n,2); //inc(i); end;
end;
procedure OutHeader(base:NativeInt); begin
writeln('Fusc numbers with more digits in base ',base,' than all previous ones:'); writeln('Value':10,'Index':10,' IndexNum/IndexNumBefore'); writeln('======':10,' =======':14);
end;
procedure CheckFuscDigits(const FF:tFusc;Base:NativeUint); var
pFF : ^tFuscElem; Dig, i,lastIdx: NativeInt;
Begin
OutHeader(base); Dig := -1; i := 0; lastIdx := 0; pFF := @FF[0];// aka FF[i] repeat //search in tight loop speeds up repeat inc(pFF); inc(i); until pFF^ >Dig;
if i>= MaxIdx then BREAK; //output write(commatize(pFF^):10,commatize(i):14);//,DIG:10); IF lastIdx> 0 then write(i/lastIdx:12:7); writeln; lastIdx := i; IF Dig >0 then Dig := Dig*Base+Base-1 else Dig := Base-1; until false; writeln;
end;
BEGIN
setlength(FuscField,MaxIdx); FuscCalc(FuscField); writeln('First 61 fusc numbers:'); OutFusc(0,60,FuscField);
CheckFuscDigits(FuscField,10); CheckFuscDigits(FuscField,11); //11 ~phi^5 1.6180..^5 = 11,09 setlength(FuscField,0); {$IFDEF WIN}readln;{$ENDIF}
END.</lang>
- Output:
First 61 fusc numbers: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Fusc numbers with more digits in base 10 than all previous ones: Value Index IndexNum/IndexNumBefore ====== ======= 1 1 11 37 37.0000000 108 1,173 31.7027027 1,076 35,499 30.2634271 10,946 699,051 19.6921322 103,682 19,573,419 27.9999871 1,010,747 615,164,587 31.4285709 Fusc numbers with more digits in base 11 than all previous ones: Value Index IndexNum/IndexNumBefore ====== ======= 1 1 11 37 37.0000000 123 1,195 32.2972973 1,364 38,229 31.9907950 15,127 1,223,339 32.0002877 167,761 39,146,837 31.9999910 1,860,498 1,252,698,795 32.0000003 real 0m1,968s user 0m1,594s sys 0m0,373s
Perl
Borrowing from the Stern-Brocot sequence task. <lang perl>use strict; use warnings; use feature 'say';
sub comma { reverse ((reverse shift) =~ s/(.{3})/$1,/gr) =~ s/^,//r }
sub stern_diatomic {
my ($p,$q,$i) = (0,1,shift); while ($i) { if ($i & 1) { $p += $q; } else { $q += $p; } $i >>= 1; } $p;
}
say "First 61 terms of the Fusc sequence:\n" . join ' ', map { stern_diatomic($_) } 0..60; say "\nIndex and value for first term longer than any previous:";
my $i = 0; my $l = -1; while ($l < 5) {
my $v = stern_diatomic($i); printf("%15s : %s\n", comma($i), comma($v)) and $l = length $v if length $v > $l; $i++;
}</lang>
- Output:
First 61 terms of the Fusc sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Index and value for first term longer than any previous: 0 : 0 37 : 11 1,173 : 108 35,499 : 1,076 699,051 : 10,946
Phix
Note that phix is 1-indexed. While there are no commas in the first 61 entries, it felt more in line with the task requirements to forego the standard comma-separated %v output.
constant limit = 20_000_000 sequence fuscs = repeat(0,limit); -- NB 1-based indexing; fusc(0)===fuscs[1] fuscs[2] = 1 -- ie fusc(1):=1 for n=3 to limit do fuscs[n] = iff(remainder(n-1,2)?fuscs[n/2]+fuscs[n/2+1]:fuscs[(n+1)/2]) end for --printf(1,"First 61 terms of the Fusc sequence:\n%v\n",{fuscs[1..61]}) string s = "" for n=1 to 61 do s&=sprintf("%,d ",fuscs[n]) end for printf(1,"First 61 terms of the Fusc sequence:\n%s\n\n",{s}) printf(1,"Elements with more digits than any previous items:\n") printf(1," Index : Value\n") integer d = 0 for n=1 to length(fuscs) do if fuscs[n]>=d then printf(1,"%,15d : %,d\n",{n-1,fuscs[n]}) d = iff(d=0?10:d*10) end if end for
- Output:
First 61 terms of the Fusc sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Elements with more digits than any previous items: Index : Value 0 : 0 37 : 11 1,173 : 108 35,499 : 1,076 699,051 : 10,946 19,573,419 : 103,682
Processing
<lang processing>void setup() {
println("First 61 terms:"); for (int i = 0; i < 60; i++) { print(fusc(i) + " "); } println(fusc(60)); println(); println("Sequence elements where number of digits of the value increase:"); int max_len = 0; for (int i = 0; i < 700000; i++) { int temp = fusc(i); if (str(temp).length() > max_len) { max_len = str(temp).length(); println("(" + i + ", " + temp + ")"); } }
}
int fusc(int n) {
if (n <= 1) { return n; } else if (n % 2 == 0) { return fusc(n / 2); } else { return fusc((n - 1) / 2) + fusc((n + 1) / 2); }
}</lang>
- Output:
First 61 terms: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Sequence elements where number of digits of the value increase: (0, 0) (37, 11) (1173, 108) (35499, 1076) (699051, 10946)
Prolog
<lang prolog>:- dynamic fusc_cache/2.
fusc(0, 0):-!. fusc(1, 1):-!. fusc(N, F):-
fusc_cache(N, F), !.
fusc(N, F):-
0 is N mod 2, !, M is N//2, fusc(M, F), assertz(fusc_cache(N, F)).
fusc(N, F):-
N1 is (N - 1)//2, N2 is (N + 1)//2, fusc(N1, F1), fusc(N2, F2), F is F1 + F2, assertz(fusc_cache(N, F)).
print_fusc_sequence(N):-
writef('First %w fusc numbers:\n', [N]), print_fusc_sequence(N, 0), nl.
print_fusc_sequence(N, M):-
M >= N, !.
print_fusc_sequence(N, M):-
fusc(M, F), writef('%w ', [F]), M1 is M + 1, print_fusc_sequence(N, M1).
print_max_fusc(N):-
writef('Fusc numbers up to %w that are longer than any previous one:\n', [N]), print_max_fusc(N, 0, 0).
print_max_fusc(N, M, _):-
M >= N, !.
print_max_fusc(N, M, Max):-
fusc(M, F), (F >= Max -> writef('n = %w, fusc(n) = %w\n', [M, F]), Max1 = max(10, Max * 10) ; Max1 = Max ), M1 is M + 1, print_max_fusc(N, M1, Max1).
main:-
print_fusc_sequence(61), print_max_fusc(1000000).</lang>
- Output:
First 61 fusc numbers: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Fusc numbers up to 1000000 that are longer than any previous one: n = 0, fusc(n) = 0 n = 37, fusc(n) = 11 n = 1173, fusc(n) = 108 n = 35499, fusc(n) = 1076 n = 699051, fusc(n) = 10946
Python
Procedural
<lang python>from collections import deque from itertools import islice, count
def fusc():
q = deque([1]) yield 0 yield 1
while True: x = q.popleft() q.append(x) yield x
x += q[0] q.append(x) yield x
def longest_fusc():
sofar = 0 for i, f in zip(count(), fusc()): if f >= sofar: yield(i, f) sofar = 10 * sofar or 10
print('First 61:')
print(list(islice(fusc(), 61)))
print('\nLength records:') for i, f in islice(longest_fusc(), 6):
print(f'fusc({i}) = {f}')
</lang>
- Output:
First 61: [0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4] Length records: fusc(0) = 0 fusc(37) = 11 fusc(1173) = 108 fusc(35499) = 1076 fusc(699051) = 10946 fusc(19573419) = 103682
Functional
By composition of pure functions, avoiding mutable variables, and confining any unavoidables to the internals of well-tested primitives: <lang python>Fusc sequence
from itertools import chain, count, islice from operator import itemgetter
- As an infinite stream of terms,
- infiniteFusc :: [Int]
def infiniteFusc():
Fusc sequence. OEIS A2487 def go(step): isEven, n, xxs = step x, xs = xxs[0], xxs[1:] if isEven: nxt = n + x return not isEven, nxt, xxs + [nxt] else: return not isEven, x, xs + [x]
return chain( [0, 1], map( itemgetter(1), iterate(go)( (True, 1, [1]) ) ) )
- Or as a function over an integer:
- fusc :: Int -> Int
def fusc(i):
Fusc sequence def go(n): if 0 == n: return (1, 0) else: x, y = go(n // 2) return (x + y, y) if 0 == n % 2 else ( x, x + y ) return 0 if 1 > i else ( go(i - 1)[0] )
- firstFuscOfEachMagnitude ::
def firstFuscOfEachMagnitude():
Non-finite stream of each term in OEIS A2487 that requires an unprecedented quantity of decimal digits. a2487 = enumerate(map(fusc, count()))
def go(e): limit = 10 ** e return next( (i, x) for i, x in a2487 if limit <= x ) return ( chain([(0, 0)], map(go, count(1))) )
- --------------------------TEST---------------------------
- main :: IO ()
def main():
Tests
print('First 61 terms:') print(showList( take(61)( map(fusc, count()) ) ))
print('\nFirst term of each decimal magnitude:') print('(Index, Term):') ixs = firstFuscOfEachMagnitude() for _ in range(0, 5): print(next(ixs))
- -------------------------GENERIC-------------------------
- iterate :: (a -> a) -> a -> Gen [a]
def iterate(f):
An infinite list of repeated applications of f to x. def go(x): v = x while True: yield v v = f(v) return lambda x: go(x)
- showList :: [a] -> String
def showList(xs):
Compact stringification of a list. return '[' + ','.join(repr(x) for x in xs) + ']'
- 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. return lambda xs: ( xs[0:n] if isinstance(xs, (list, tuple)) else list(islice(xs, n)) )
- MAIN ---
if __name__ == '__main__':
main()</lang>
- Output:
First 61 terms: [0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4] First term of each decimal magnitude: (Index, Term): (0, 0) (37, 11) (1173, 108) (35499, 1076) (699051, 10946)
Quackery
<lang Quackery>
[ 1 & ] is odd ( n --> b )
[ 0 swap [ dip 1+ 10 / dup 0 = until ] drop ] is digits ( n --> n )
[ dup dup size dup odd iff [ dup 1 - 2 / dip [ 1 + 2 / peek over ] peek + ] else [ 2 / peek ] join ] is nextfusc ( [ --> [ )
say "First 61 terms." cr ' [ 0 1 ] 59 times nextfusc witheach [ echo sp ] cr cr say "Terms where the digit count increases." cr say "fusc(0) = 0" cr 1 ' [ 0 1 ] [ nextfusc dup -1 peek digits rot 2dup > iff [ drop swap say "fusc(" dup -1 peek echo say ") = " dup size 1 - echo cr ] else [ nip swap ] dup size 1000000 = until ] 2drop</lang>
- Output:
First 61 terms. 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Terms where the digit count increases. fusc(0) = 0 fusc(11) = 37 fusc(108) = 1173 fusc(1076) = 35499 fusc(10946) = 699051
R
I believe that this code demonstrates a great truth of R: It is amazing with numbers, but terrible with strings. There is really no good reason why checking how long a number is and printing it nicely should be hardest parts of this task.
Our first step is to adapt the 0-indexed definition to our 1-indexed language, letting us complete the first task. <lang r>firstNPlus1FuscNumbers<-function(n) {
if(n==0){return(0)}else{fusc<-c(0,1)} for(i in (2+seq_len(n-1))) { if(i %% 2 == 0){fusc[i]<-fusc[i/2]+fusc[(i+2)/2]} else{fusc[i]<-fusc[(i+1)/2]} } fusc
} first61<-firstNPlus1FuscNumbers(60) print(first61)</lang> The second task's requirements are somewhat strange. It asks for the number, implying that there is only one, but it is clear that there are several. If we only want the first such number, then the task is trivial. As we have already seen it in the n=60 output, we don't even have to be smart. Indeed, if we were being smart, we'd say that the answer was trivial: 0 at index 1.
A proper solution that only gives one non-trivial result is as follows: <lang r>index<-which.max(nchar(first61)==2) number<-first61[index] cat("The first fusc number that is longer than all previous fusc numbers is",number,
"and it occurs at index",index)</lang>
Regardless, as many of the other solutions have displayed many such numbers (e.g. the 6 digit case), we will do the same. This complicates matters in some unexpected ways. For example, nchar misbehaves once its inputs get large enough for R to default to scientific notation. One nice solution is to use format, which also allows us to add the required commas: <lang r>twentyMillion<-firstNPlus1FuscNumbers(2*10^7-1) twentyMillionCountable<-format(twentyMillion,scientific = FALSE,trim = TRUE) indices<-sapply(2:6, function(x) which.max(nchar(twentyMillionCountable)==x)) numbers<-twentyMillion[indices] cat("Some fusc numbers that are longer than all previous fusc numbers are:\n",
paste0(format(twentyMillion[indices],scientific = FALSE,trim = TRUE,big.mark = ","), " (at index ",format(indices,trim = TRUE,big.mark = ","),")\n"))</lang>
- Output:
> print(first61) [1] 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 [38] 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 > cat("The first fusc number that is longer than all previous fusc numbers is",number, + "and it occurs at index",index) The first fusc number that is longer than all previous fusc numbers is 11 and it occurs at index 38 > cat("Some fusc numbers that are longer than all previous fusc numbers are:\n", + paste0(format(twentyMillion[indices],scientific = FALSE,trim = TRUE,big.mark = ","), + " (at index ",format(indices,trim = TRUE,big.mark = ","),")\n")) Some fusc numbers that are longer than all previous fusc numbers are: 11 (at index 38) 108 (at index 1,174) 1,076 (at index 35,500) 10,946 (at index 699,052) 103,682 (at index 19,573,420)
Racket
<lang racket>#lang racket
(require racket/generator)
(define (memoize f)
(define table (make-hash)) (λ args (hash-ref! table args (thunk (apply f args)))))
(define fusc
(memoize (λ (n) (cond [(<= n 1) n] [(even? n) (fusc (/ n 2))] [else (+ (fusc (/ (sub1 n) 2)) (fusc (/ (add1 n) 2)))]))))
(define (comma x)
(string-join (reverse (for/list ([digit (in-list (reverse (string->list (~a x))))] [i (in-naturals)]) (cond [(and (= 0 (modulo i 3)) (> i 0)) (string digit #\,)] [else (string digit)]))) ""))
- Task 1
(displayln (string-join (for/list ([i (in-range 61)]) (comma (fusc i))) " ")) (newline)
- Task 2
(define gen
(in-generator (let loop ([prev 0] [i 0]) (define result (fusc i)) (define len (string-length (~a result))) (cond [(> len prev) (yield (list i result)) (loop len (add1 i))] [else (loop prev (add1 i))]))))
(for ([i (in-range 5)] [x gen])
(match-define (list index result) x) (printf "~a: ~a\n" (comma index) (comma result)))</lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 0: 0 37: 11 1,173: 108 35,499: 1,076 699,051: 10,946
Raku
(formerly Perl 6)
Recurrence
<lang perl6>my @Fusc = 0, 1, 1, { |(@Fusc[$_ - 1] + @Fusc[$_], @Fusc[$_]) given ++$+1 } ... *;
sub comma { $^i.flip.comb(3).join(',').flip }
put "First 61 terms of the Fusc sequence:\n{@Fusc[^61]}" ~
"\n\nIndex and value for first term longer than any previous:";
for flat 'Index', 'Value', 0, 0, (1..4).map({
my $l = 10**$_; @Fusc.first(* > $l, :kv).map: &comma }) -> $i, $v { printf "%15s : %s\n", $i, $v
}</lang>
- Output:
First 61 terms of the Fusc sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Index and value for first term longer than any previous: Index : Value 0 : 0 37 : 11 1,173 : 108 35,499 : 1,076 699,051 : 10,946
Recursive
Alternative version using Raku's multiple-dispatch feature. This version is significantly slower than the one above, but it's definitely prettier. <lang perl6> multi fusc( 0 ) { 0 } multi fusc( 1 ) { 1 } multi fusc( $n where $n %% 2 ) { fusc $n div 2 } multi fusc( $n ) { [+] map *.&fusc, ( $n - 1 ) div 2, ( $n + 1 ) div 2 } put map *.&fusc, 0..60;</lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
REXX
version 1, standard formatting
<lang rexx>/*REXX program calculates and displays the fusc (or Stern's Diatomic) sequence. */ parse arg st # xw . /*obtain optional arguments from the CL*/ if st== | st=="," then st= 0 /*Not specified? Then use the default.*/ if #== | #=="," then #= 61 /* " " " " " " */ if xw== | xw=="," then xw= 0 /* " " " " " " */ list= xw<1 /*boolean value: LIST to show numbers*/ @.=; @.0= 0; @.1= 1 /*assign array default; assign low vals*/ mL= 0 /*the maximum length (digits) so far. */ $= /* " list of fusc numbers " " */
do j=0 for # /*process a bunch of integers from zero*/ if j>1 then if j//2 then do; _= (j-1) % 2; p= (j+1) % 2; @.j= @._ + @.p; end else do; _= j % 2; @.j= @._; end if list then if j>=st then $= $ commas(@.j) /*add it to a list*/ else nop /*NOP≡placeholder.*/ else do; if length(@.j)<=mL then iterate /*still too small.*/ mL= length(@.j) /*found increase. */ if mL==1 then say '═══index═══ ═══fusc number═══' say right( commas(j), 9) right( commas(@.j), 14) if mL==xw then leave /*Found max length? Then stop looking.*/ end /* [↑] display fusc #s of maximum len.*/ end /*j*/ /*$ has a superfluous leading blank. */
if $\== then say strip($) /*display a horizontal list of fusc #s.*/ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg ?; do _=length(?)-3 to 1 by -3; ?=insert(',', ?, _); end; return ?</lang>
- output when using the default inputs:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
- output when using the default inputs: , 999999999 5
═══index═══ ═══fusc number═══ 0 0 37 11 1,173 108 35,499 1,076 699,051 10,946
version 2, formatted with rows re─starting whenever a 1 (unity) appears
<lang rexx>/*REXX program calculates and displays the fusc (or Stern's Diatomic) sequence. */ parse arg st # xw . /*obtain optional arguments from the CL*/ if st== | st=="," then st= 0 /*Not specified? Then use the default.*/ if #== | #=="," then #= 256 /* " " " " " " */ if xw== | xw=="," then xw= 0 /* " " " " " " */ list= xw<1 /*boolean value: LIST to show numbers*/ @.=; @.0= 0; @.1= 1 /*assign array default; assign low vals*/ mL= 0 /*the maximum length (digits) so far. */ $= /* " list of fusc numbers " " */
do j=0 for # /*process a bunch of integers from zero*/ if j>1 then if j//2 then do; _= (j-1) % 2; p= (j+1) % 2; @.j= @._ + @.p; end else do; _= j % 2; @.j= @._; end if list then if j>=st then $= $ commas(@.j) /*add it to a list*/ else nop /*NOP≡placeholder.*/ else do; if length(@.j)<=mL then iterate /*still too small.*/ mL= length(@.j) /*found increase. */ if mL==1 then say '═══index═══ ═══fusc number═══' say right( commas(j), 9) right( commas(@.j), 14) if mL==xw then leave /*Found max length? Then stop looking.*/ end /* [↑] display fusc #s of maximum len.*/ end /*j*/ /*$ has a superfluous leading blank. */
if $== then exit 0 /*display a horizontal list of fusc #s.*/ row= -1 /*output will be starting ar row zero.*/ $$= 0 /*initialize with the zeroth entry (=0)*/
do k=2 for #; y= word($, k) /*start processing with the 2nd number.*/ if y==1 then do; row= row + 1 /*Is it unity? Then bump row number.*/ say 'row('row")=" $$ /*display the row that was just created*/ $$= 1 /*initialize a new row with 1 (unity).*/ end else $$= $$ y /*Not unity? Just append it to a row.*/ end /*k*/
if $$\== then say "row("row+1')=' $$ /*display any residual data in the row.*/ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg ?; do _=length(?)-3 to 1 by -3; ?=insert(',', ?, _); end; return ?</lang>
- output when using the default inputs:
(Shown at 70% size.)
row(0)= 0 row(1)= 1 row(2)= 1 2 row(3)= 1 3 2 3 row(4)= 1 4 3 5 2 5 3 4 row(5)= 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 row(6)= 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 9 5 6 row(7)= 1 7 6 11 5 14 9 13 4 15 11 18 7 17 10 13 3 14 11 19 8 21 13 18 5 17 12 19 7 16 9 11 2 11 9 16 7 19 12 17 5 18 13 21 8 19 11 14 3 13 10 17 7 18 11 15 4 13 9 14 5 11 6 7 row(8)= 1 8 7 13 6 17 11 16 5 19 14 23 9 22 13 17 4 19 15 26 11 29 18 25 7 24 17 27 10 23 13 16 3 17 14 25 11 30 19 27 8 29 21 34 13 31 18 23 5 22 17 29 12 31 19 26 7 23 16 25 9 20 11 13 2 13 11 20 9 25 16 23 7 26 19 31 12 29 17 22 5 23 18 31 13 34 21 29 8 27 19 30 11 25 14 17 3 16 13 23 10 27 17 24 7 25 18 29 11 26 15 19 4 17 13 22 9 23 14 19 5 16 11 17 6 13 7 8
- Output observation: note that each (positive) row doubles in size (number of entries), and starts with unity (1), and
- ends with the number of the row (if the number of sequence elements is a power of two).
Ring
<lang ring>
- Project: Fusc sequence
max = 60 fusc = list(36000) fusc[1] = 1 see "working..." + nl see "wait for done..." + nl see "The first 61 fusc numbers are:" + nl fuscseq(max) see "0" for m = 1 to max
see " " + fusc[m]
next
see nl see "The fusc numbers whose length > any previous fusc number length are:" + nl see "Index Value" + nl see " 0 0" + nl d = 10 for i = 1 to 36000
if fusc[i] >= d see " " + i + " " + fusc[i] + nl if d = 0 d = 1 ok d = d*10 ok
next see "done..." + nl
func fuscseq(max)
for n = 2 to 36000 if n%2 = 1 fusc[n] = fusc[(n-1)/2] + fusc[(n+1)/2] but n%2 = 0 fusc[n] = fusc[n/2] ok next
</lang>
- Output:
working... wait for done... The first 61 fusc numbers are: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 The fusc numbers whose length > any previous fusc number length are: Index Value 0 0 37 11 1173 108 35499 1076 done...
Ruby
Using two Enumerators; the second making use of the first: <lang ruby>fusc = Enumerator.new do |y|
y << 0 y << 1 arr = [0,1] 2.step do |n| res = n.even? ? arr[n/2] : arr[(n-1)/2] + arr[(n+1)/2] y << res arr << res end
end
fusc_max_digits = Enumerator.new do |y|
cur_max, cur_exp = 0, 0 0.step do |i| f = fusc.next if f >= cur_max cur_exp += 1 cur_max = 10**cur_exp y << [i, f] end end
end
puts fusc.take(61).join(" ") fusc_max_digits.take(6).each{|pair| puts "%15s : %s" % pair } </lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 0 : 0 11 : 37 108 : 1173 1076 : 35499 10946 : 699051 103682 : 19573419
Rust
<lang rust>fn fusc_sequence() -> impl std::iter::Iterator<Item = u32> {
let mut sequence = vec![0, 1]; let mut n = 0; std::iter::from_fn(move || { if n > 1 { sequence.push(match n % 2 { 0 => sequence[n / 2], _ => sequence[(n - 1) / 2] + sequence[(n + 1) / 2], }); } let result = sequence[n]; n += 1; Some(result) })
}
fn main() {
println!("First 61 fusc numbers:"); for n in fusc_sequence().take(61) { print!("{} ", n) } println!();
let limit = 1000000000; println!( "Fusc numbers up to {} that are longer than any previous one:", limit ); let mut max = 0; for (index, n) in fusc_sequence().take(limit).enumerate() { if n >= max { max = std::cmp::max(10, max * 10); println!("index = {}, fusc number = {}", index, n); } }
}</lang>
- Output:
First 61 fusc numbers: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Fusc numbers up to 1000000000 that are longer than any previous one: index = 0, fusc number = 0 index = 37, fusc number = 11 index = 1173, fusc number = 108 index = 35499, fusc number = 1076 index = 699051, fusc number = 10946 index = 19573419, fusc number = 103682 index = 615164587, fusc number = 1010747
Sidef
<lang ruby>func fusc(n) is cached {
return 0 if n.is_zero return 1 if n.is_one
n.is_even ? fusc(n/2) : (fusc((n-1)/2) + fusc(((n-1)/2)+1))
}
say ("First 61 terms of the Stern-Brocot sequence: ", 61.of(fusc).join(' '))
say "\nIndex and value for first term longer than any previous:" printf("%15s : %s\n", "Index", "Value");
var (index=0, len=0)
5.times {
index = (index..Inf -> first_by { fusc(_).len > len }) len = fusc(index).len printf("%15s : %s\n", index.commify, fusc(index).commify)
}</lang>
- Output:
First 61 terms of the Stern-Brocot sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Index and value for first term longer than any previous: Index : Value 0 : 0 37 : 11 1,173 : 108 35,499 : 1,076 699,051 : 10,946
Swift
<lang swift>struct FuscSeq: Sequence, IteratorProtocol {
private var arr = [0, 1] private var i = 0
mutating func next() -> Int? { defer { i += 1 }
guard i > 1 else { return arr[i] }
switch i & 1 { case 0: arr.append(arr[i / 2]) case 1: arr.append(arr[(i - 1) / 2] + arr[(i + 1) / 2]) case _: fatalError() }
return arr.last! }
}
let first = FuscSeq().prefix(61)
print("First 61: \(Array(first))")
var max = -1
for (i, n) in FuscSeq().prefix(20_000_000).enumerated() {
let f = String(n).count
if f > max { max = f
print("New max: \(i): \(n)") }
}</lang>
- Output:
First 61: [0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4] New max: 0: 0 New max: 37: 11 New max: 1173: 108 New max: 35499: 1076 New max: 699051: 10946 New max: 19573419: 103682
Tcl
<lang Tcl>proc fusc n {
if {$n < 2} { return $n }
if {[info exists ::g_fusc($n)]} { return $::g_fusc($n) }
if {$n % 2} { ;# n is odd set r [expr {[fusc [expr {($n-1)/2}]] + [fusc [expr {($n+1)/2}]]}] } else { ;# n is even set r [fusc [expr {$n/2}]] }
if {$n < 999999} { set ::g_fusc($n) $r }
return $r
}
proc ,,, {str {sep ,} {grouplen 3}} {
set strlen [string length $str] set padlen [expr {($grouplen - ($strlen % $grouplen)) % $grouplen}] set r [regsub -all ... [string repeat " " $padlen]$str &$sep] return [string range $r $padlen end-[string length $sep]]
}
proc tabline {a b c} {
puts "[format %2s $a] [format %10s $b] [format %8s $c]"
}
proc doit Template:Nmax 20000000 {
for {set i 0} {$i < 61} {incr i} { puts -nonewline " [fusc $i]" } puts "" tabline L n fusc(n) set maxL 0 for {set n 0} {$n < $nmax} {incr n} { set f [fusc $n] set L [string length $f] if {$L > $maxL} { set maxL $L tabline $L [,,, $n] [,,, $f] } }
} doit</lang>
- Output:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 L n fusc(n) 1 0 0 2 37 11 3 1,173 108 4 35,499 1,076 5 699,051 10,946 6 19,573,419 103,682 real 2m5.559s
Vala
<lang vala>int fusc(int n) {
if (n == 0 || n == 1) return n; else if (n % 2 == 0) return fusc(n / 2); else return fusc((n - 1) / 2) + fusc((n + 1) / 2);
}
void main() {
print("The first 61 fusc numbers:\n"); for (int i = 0; i < 61; i++) print(@"$(fusc(i)) "); print("\n\nThe fusc numbers whose lengths are greater than those of previous fusc numbers:\n"); print(" n fusc(n)\n"); print("-------------------\n"); var max_length = 0; for (int i = 0; i < 700000; i++) { var f = fusc(i); var length = f.to_string().length; if (length > max_length) { max_length = length; print("%9d %9d\n", i, f); } }
}</lang>
- Output:
The first 61 fusc numbers: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 The fusc numbers whose lengths are greater than those of previous fusc numbers: n fusc(n) ------------------- 0 0 37 11 1173 108 35499 1076 699051 10946
Visual Basic .NET
<lang vbnet>Module Module1
Dim n As Integer = 61, l As List(Of Integer) = {0, 1}.ToList
Function fusc(n As Integer) As Integer If n < l.Count Then Return l(n) fusc = If((n And 1) = 0, l(n >> 1), l((n - 1) >> 1) + l((n + 1) >> 1)) l.Add(fusc) End Function
Sub Main(args As String()) Dim lst As Boolean = True, w As Integer = -1, c As Integer = 0, fs As String = "{0,11:n0} {1,-9:n0}", res As String = "" Console.WriteLine("First {0} numbers in the fusc sequence:", n) For i As Integer = 0 To Integer.MaxValue Dim f As Integer = fusc(i) If lst Then If i < 61 Then Console.Write("{0} ", f) Else lst = False Console.WriteLine() Console.WriteLine("Points in the sequence where an item has more digits than any previous items:") Console.WriteLine(fs, "Index\", "/Value") : Console.WriteLine(res) : res = "" End If End If Dim t As Integer = f.ToString.Length If t > w Then w = t res &= If(res = "", "", vbLf) & String.Format(fs, i, f) If Not lst Then Console.WriteLine(res) : res = "" c += 1 : If c > 5 Then Exit For End If Next : l.Clear() End Sub
End Module </lang>
- Output:
First 61 numbers in the fusc sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Points in the sequence where an item has more digits than any previous items: Index\ /Value 0 0 37 11 1,173 108 35,499 1,076 699,051 10,946 19,573,419 103,682
Wren
<lang ecmascript>import "/fmt" for Fmt
System.print("The first 61 numbers in the fusc sequence are:") var fusc = [0, 1] var fusc2 = 0, 0 var maxLen = 1 var n = 2 while (n < 20e6) { // limit to indices under 20 million say
var f = (n % 2 == 0) ? fusc[n/2] : fusc[(n-1)/2] + fusc[(n+1)/2] fusc.add(f) var len = "%(f)".count if (len > maxLen) { maxLen = len if (n <= 60) { fusc2.add([n, f]) } else { System.print("%(Fmt.dc(10, n)) %(Fmt.dc(0, f))") } } if (n == 60 ) { for (f in fusc) System.write("%(f) ") System.print("\n\nFirst terms longer than any previous ones for indices < 20,000,000:") System.print(" Index Value") for (iv in fusc2) System.print("%(Fmt.d(10, iv[0])) %(iv[1])") } n = n + 1
}</lang>
- Output:
The first 61 numbers in the fusc sequence are: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 First terms longer than any previous ones for indices < 20,000,000: Index Value 0 0 37 11 1,173 108 35,499 1,076 699,051 10,946 19,573,419 103,682
zkl
<lang zkl>fuscs:=List.createLong(1_000_000, 0); fuscs[1]=1; // we'll just use a big count foreach n in ([2..fuscs.len()-1]){ // and generate
fuscs[n]=( if(n.isEven()) fuscs[n/2] else fuscs[(n-1)/2] + fuscs[(n+1)/2] )
}
println("First 61 terms of the Stern-Brocot sequence:"); fuscs[0,61].concat(" ").println();
println("\nIndex and value for first term longer than any previous:"); println(" Index : Value"); prevMax:=-1; foreach n in (fuscs.len()){
f,fd := fuscs[n], f.numDigits; if(fd>prevMax){ println("%15,d : %,d".fmt(n,f)); prevMax=fd }
}</lang>
- Output:
First 61 terms of the Stern-Brocot sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 Index and value for first term longer than any previous: Index : Value 0 : 0 37 : 11 1,173 : 108 35,499 : 1,076 699,051 : 10,946