Fusc sequence

Definitions

The fusc integer sequence is defined as:

fusc(0) = 0
fusc(1) = 1
for n>1, the n^th 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

RosettaCode Stern-Brocot sequence

Also see

the MathWorld entry: Stern's Diatomic Series.
the OEIS entry: A2487.

11l

Translation of: Kotlin

<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

Translation of: Nim

<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

AutoHotkey

<lang AutoHotkey>fusc:=[], fusc[0]:=0, fusc[1]:=1, n:=1, l:=0, result:=""

while (StrLen(fusc[n]) < 5)

   fusc[++n] := Mod(n, 2) ? fusc[floor((n-1)/2)] + fusc[Floor((n+1)/2)] : fusc[floor(n/2)]

while (A_Index <= 61)

   result .= (result = "" ? "" : ",") fusc[A_Index-1]

result .= "`n`nfusc number whose length is greater than any previous fusc number length:`nindex`tnumber`n" for i, v in fusc

   if (l < StrLen(v))
       l := StrLen(v), result .= i "`t" v "`n"

MsgBox % 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
fusc number whose length is greater than any previous fusc number length:
index	number
0	0
37	11
1173	108
35499	1076
699051	10946

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

•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

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++

Translation of: 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

CLU

Translation of: Python

<lang clu>fusc = iter () yields (int)

   q: array[int] := array[int]$[1]
   yield(0)
   yield(1)
   
   while true do
       x: int := array[int]$reml(q)
       array[int]$addh(q,x)
       yield(x)
       
       x := x + array[int]$bottom(q)
       array[int]$addh(q,x)
       yield(x)
   end

end fusc

longest_fusc = iter () yields (int,int)

   sofar: int := 0
   count: int := 0
   
   for f: int in fusc() do
       if f >= sofar then
           yield (count,f)
           sofar := 10*sofar
           if sofar=0 then sofar:=10 end
       end
       count := count + 1
   end

end longest_fusc

start_up = proc ()

   po: stream := stream$primary_output()
   
   stream$putl(po, "First 61:")
   n: int := 0
   for f: int in fusc() do
       stream$puts(po, int$unparse(f) || " ")
       n := n + 1
       if n = 61 then break end
   end
   
   stream$putl(po, "\nLength records:")
   n := 0
   for i, f: int in longest_fusc() do
       stream$putl(po, "fusc(" || int$unparse(i) || ") = " || int$unparse(f))
       n := n + 1
       if n = 5 then break end
   end

end start_up</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

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!"\nFusc numbers with more digits than any previous (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:

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 any previous (1 to 5 digits):
fusc(0)=0
fusc(37)=11
fusc(1173)=108
fusc(35499)=1076
fusc(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;
 
 format!"First %d fusc numbers: "(N_FIRST).write;
 foreach( n; 0..N_FIRST ) n.fusc.format!"%d ".write; 
 writeln; 
     
 format!"\nFusc numbers with more digits than any previous (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:

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 any previous (1 to 6 digits):
fusc(0)=0
fusc(37)=11
fusc(1173)=108
fusc(35499)=1076
fusc(699051)=10946
fusc(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); " ";

   If f(i) >= d Then
       Print Using "###########," ; i; f(i)
       If d = 0 Then d = 1
       d *= 10
   End If

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

Translation of: Java

<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

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

Translation of: Python

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)

jq

Adapted from Wren

Works with: jq

Works with gojq, the Go implementation of jq

Preliminaries <lang jq># input should be a non-negative integer def commatize:

 # "," is 44
 def digits: tostring | explode | reverse;
 [foreach digits[] as $d (-1; .+1;
    (select(. > 0 and . % 3 == 0)|44), $d)]
 | reverse | implode  ;

def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;</lang> Fusc Sequence <lang jq># Save space by truncating the beginning of the array def fusc:

  0, 1,
  foreach range(2; infinite) as $n ([0, 1];
     ($n % 2 == 0) as $even
     | if $even then . + [.[1]] else.[1:] + [.[1] + .[2]] end;
     .[-1] );

Report first longest

def fusc( $mx ):

 def l: commatize|lpad(10);

 foreach limit( $mx; fusc ) as $f ({ maxLen: 0, n: 0 };
   .emit = false
   | ("\($f)"|length) as $len
   | if $len > .maxLen
     then .maxLen = $len
     | .emit = "\(.n|l)  \($f|commatize)"
     else .
     end
     | .n += 1
     ;
     select(.emit).emit
   );

First $first numbers in the fusc sequence

61 as $first | 2e6 as $mx | "The first \($first) numbers in the fusc sequence are:",

  ([limit($first; fusc)]| map(tostring) | join(" ")) ,

  "\nFirst terms longer than any previous ones for indices < \($mx + 0 |commatize):",
  "     Index  Value",
  fusc($mx)</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

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

Translation of: Go

<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

Translation of: C

<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)

Translation of: Python

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

Works with: Free 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

Works with: SWI 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

 [ 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 rsplus>firstNFuscNumbers <- function(n) {

 stopifnot(n > 0)
 if(n == 1) return(0) else fusc <- c(0, 1)
 if(n > 2)
 {
   for(i in seq(from = 3, to = n, by = 1))
   {
     fusc[i] <- if(i %% 2) fusc[(i + 1) / 2] else fusc[i / 2] + fusc[(i + 2) / 2]
   }
 }
 fusc

} first61 <- firstNFuscNumbers(61) cat("The first 61 Fusc numbers are:", "\n", first61, "\n")</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=61 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 rsplus>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, "\n")</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 rsplus>twentyMillion <- firstNFuscNumbers(2 * 10^7) 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:

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 first fusc number that is longer than all previous fusc numbers is 11 and it occurs at index 38 
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

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]

   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

Translation of: Nim

<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

Translation of: C#

<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

Library: Wren-fmt

<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