Fusc sequence

From Rosetta Code
Task
Fusc sequence
You are encouraged to solve this task according to the task description, using any language you may know.


Definitions

The   fusc   integer sequence is defined as:

  •   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



11l[edit]

Translation of: Kotlin
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]))
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[edit]

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;
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[edit]

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
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[edit]

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}
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:

-- 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
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[edit]

Translation of: Nim
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
        ]
    ]
]
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[edit]

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
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[edit]

# 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)
}
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

BASIC256[edit]

global f, max
max = 36000
dim f(max)

call fusc()

for i = 0 to 60
    print f[i]; " ";
next i

print : print
print "     Index         Value"
d = 0
for i = 0 to max-1
    if f[i] >= d then
	print rjust(string(i),10," "), rjust(string(f[i]),10," ")
	if d = 0 then d = 1
	d *= 10
    end if
next i
end

subroutine fusc()
    f[0] = 0 : f[1] = 1
    for n = 2 to max-1
	if (n mod 2) then
	    f[n] = f[(n-1)/2] + f[(n+1)/2]
	else
	    f[n] = f[n/2]
	end if
    next n
end subroutine

BQN[edit]

Works in: CBQN

Fusc computes fusc numbers iteratively.

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
⟨ 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
                   ┘

C[edit]

#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;
}

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#[edit]

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();
    }
}
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++[edit]

Translation of: C#
#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;
}
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[edit]

Translation of: Python
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
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[edit]

Built-in memoization[edit]

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;
    } 
}
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[edit]

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;
    } 
}
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[edit]

See Pascal.

Dyalect[edit]

Translation of: C#
let n = 61
let l = [0, 1]

func fusc(n) {
    return l[n] when n < l.Length()
    let f = (n &&& 1) == 0 ? l[n >>> 1] : l[(n - 1) >>> 1] + l[(n + 1) >>> 1]
    l.Add(f)
    return f
}

var lst = true
var w = -1
var c = 0
var t = nil
var res = ""

print("First \(n) numbers in the fusc sequence:")
for i in 0..Integer.Max {
    let f = fusc(i)
    if lst {
        if i < 61 {
            print("\(f) ", terminator: "")
        } else {
            lst = false
            print("")
            print("Points in the sequence where an item has more digits than any previous items:")
            print("Index/Value:")
            print(res)
            res = ""
        }
    }
    t = f.ToString().Length()
    if t > w {
        w = t
        res += (res == "" ? "" : "\n") + "\(i)/\(f)"
        if !lst {
            print(res)
            res = ""
        }
        c += 1
        if c > 5 {
            break
        }
    }
}
l.Clear()
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

F#[edit]

The Function[edit]

// 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))

The Tasks[edit]

Print first 62 elements
fusc |> Seq.take 61 |> Seq.iter(fun(_,g)->printf "%d " g); printfn ""
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

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)
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[edit]

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
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


Forth[edit]

\ Gforth 0.7.9_20211014

: fusc ( n -- n)                     \  input n -- output fusc(n)
  dup  dup  0= swap  1 = or          \  n = 0 or 1
  if  exit                           \  return n
  else dup 2 mod 0=                  \  test even
       if 2/ recurse                 \  even fusc(n)= fusc(n/2)
       else dup  1- 2/ recurse       \  odd  fusc(n) = fusc((n-1)/2) +
            swap 1+ 2/ recurse  +    \                 fusc((n+1)/2)
       then
  then
;

: cntDigits ( n -- n )               \ returns the numbers of digits
  0 begin 1+ swap
        10 /
        swap  over
  0= until
  swap drop
;

: fuscLen ( n -- )                    \ count until end index
  cr 1   swap  0
  do
    i fusc cntDigits
    over > if 1+
                  ." fusc( " i . ." ) : "
                  i fusc  . cr
           then
  loop
;

: firstFusc ( n -- )                  \ show  fusc(i)   until  limit
   dup ." First " . ." fusc(n) : " cr
   0 do  I fusc .  loop cr
;

61 firstFusc

20 1000 1000 * * fuscLen

bye
Output:
First 61 fusc(n) : 
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( 37 ) : 11 
fusc( 1173 ) : 108 
fusc( 35499 ) : 1076 
fusc( 699051 ) : 10946 
fusc( 19573419 ) : 103682

FreeBASIC[edit]

' 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
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[edit]

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[edit]

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]))
    }
}
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[edit]

Translation of: Java
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]
        }
    }
}
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[edit]

---------------------- 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)
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:

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
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[edit]

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
┌─────┬─┬──┬────┬─────┐
index037117335499
├─────┼─┼──┼────┼─────┤
value011 108 1076
└─────┴─┴──┴────┴─────┘

Java[edit]

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]);
        }
    }
}
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[edit]

Functional[edit]

Translation of: Python

A composition of pure generic functions:

(() => {
    "use strict";

    // ---------------------- FUSC -----------------------

    // fusc :: Int -> Int
    const fusc = i => {
        const go = n =>
            0 === n ? [
                1, 0
            ] : (() => {
                const [x, y] = go(Math.floor(n / 2));

                return 0 === n % 2 ? (
                    [x + y, y]
                ) : [x, x + y];
            })();

        return 1 > i ? (
            0
        ) : go(i - 1)[0];
    };


    // ---------------------- TEST -----------------------
    const main = () => {
        const terms = enumFromTo(0)(60).map(fusc);

        return [
                "First 61 terms:",
                `[${terms.join(",")}]`,
                "",
                "(Index, Value):",
                firstWidths(5).reduce(
                    (a, x) => [x.slice(1), ...a],
                    []
                )
                .map(([i, x]) => `(${i}, ${x})`)
                .join("\n")
            ]
            .join("\n");
    };


    // firstWidths :: Int -> [(Int, Int)]
    const firstWidths = n => {
        const nxtWidth = xs => {
            const
                fi = fanArrow(fusc)(x => x),
                [w, i] = xs[0],
                [x, j] = Array.from(
                    until(
                        v => w <= `${v[0]}`.length
                    )(
                        v => fi(1 + v[1])
                    )(fi(i))
                );

            return [
                [1 + w, j, x],
                ...xs
            ];
        };

        return until(x => n < x[0][0])(
            nxtWidth
        )([
            [2, 0, 0]
        ]);
    };


    // ---------------- GENERIC FUNCTIONS ----------------

    // enumFromTo :: Int -> Int -> [Int]
    const enumFromTo = m =>
        n => Array.from({
            length: 1 + n - m
        }, (_, i) => m + i);


    // fanArrow (&&&) ::
    // (a -> b) -> (a -> c) -> (a -> (b, c))
    const fanArrow = f =>
        // A combined function, given f and g,
        // from x to a tuple of (f(x), g(x))
        // ((,) . f <*> g)
        g => x => [f(x), g(x)];


    // until :: (a -> Bool) -> (a -> a) -> a -> a
    const until = p =>
        // The value resulting from successive applications
        // of f to f(x), starting with a seed value x,
        // and terminating when the result returns true
        // for the predicate p.
        f => {
            const go = x =>
                p(x) ? x : go(f(x));

            return go;
        };

    // MAIN ---
    return main();
})();
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[edit]

Adapted from Wren

Works with: jq

Works with gojq, the Go implementation of jq

Preliminaries

# 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] + .;

Fusc Sequence

# 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)
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[edit]

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()
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[edit]

Translation of: Go
// 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)
    }
}
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[edit]

Translation of: C
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()
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[edit]

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 = {{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
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[edit]

Using recursive procedure[edit]

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.

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}"
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)[edit]

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.

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
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[edit]

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

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.
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[edit]

Borrowing from the Stern-Brocot sequence task.

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++;
}
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[edit]

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

Picat[edit]

main =>
  println("First 61 fusc numbers:"),
  println([fusc(I) : I in 0..60]),
  nl,
  println("Points in the sequence whose length is greater than any previous fusc number length:\n"),
  println("   Index     fusc  Len"),  
  largest_fusc_string(20_000_000).

table
fusc(0) = 0.
fusc(1) = 1.
fusc(N) = fusc(N//2), even(N) => true.
fusc(N) = fusc((N-1)//2) + fusc((N+1)//2) => true.

largest_fusc_string(Limit) =>
  largest_fusc_string(0,Limit,0).

largest_fusc_string(Limit,Limit,_). 
largest_fusc_string(N,Limit,LargestLen) :-
  N <= Limit,
  F = fusc(N),
  Len = F.to_string.len,
  (Len > LargestLen ->
     printf("%8d %8d %4d\n",N,F,Len),     
     LargestLen1 = Len
   ;
     LargestLen1 = LargestLen
  ),
  largest_fusc_string(N+1,Limit,LargestLen1).
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]

Points in the sequence whose length is greater than any previous fusc number length:

   Index     fusc  Len
       0        0    1
      37       11    2
    1173      108    3
   35499     1076    4
  699051    10946    5
19573419   103682    6

Processing[edit]

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);
  }
}
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[edit]

Works with: SWI 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).
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[edit]

Procedural[edit]

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}')
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[edit]

By composition of pure functions, avoiding mutable variables, and confining any unavoidables to the internals of well-tested primitives:

'''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()
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[edit]

  [ 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
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[edit]

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.

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")

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:

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")

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:

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"))
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[edit]

#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)))
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[edit]

(formerly Perl 6)

Recurrence[edit]

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
}
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[edit]

Alternative version using Raku's multiple-dispatch feature. This version is significantly slower than the one above, but it's definitely prettier.

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;
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[edit]

version 1, standard formatting[edit]

/*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 ?
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[edit]

/*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 ?
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[edit]

# 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
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[edit]

Using two Enumerators; the second making use of the first:

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 }
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[edit]

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);
        }
    }
}
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[edit]

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)
}
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[edit]

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)")
  }
}
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[edit]

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 {{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
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

uBasic/4tH[edit]

Translation of: C

Only numbers up to 35500 are listed, otherwise it would take an unreasonable amount of time to run this program.

Print "Index-------Value"

For i = 0 To 60
  Print Using "____#"; i; Using "___________#"; FUNC(_fusc(i))
Next

Proc _printLargeFuscs (35500)
End

_fusc
  Param (1)
  
  If (a@ = 0) + (a@ = 1) Then Return (a@)
  If (a@ % 2) = 0 Then Return (FUNC(_fusc(a@/2)))
Return (FUNC(_fusc((a@ - 1)/2)) + FUNC(_fusc((a@ + 1)/2)))
 
_printLargeFuscs
  Param (1)
  Local (4)             '              (int) i, f, len, maxLen = 1
  
  e@ = 1
  Print "\n\nPrinting all largest Fusc numbers upto "; a@; "\nIndex-------Value"
 
  For b@ = 0 To a@
    c@ = FUNC(_fusc(b@))
    d@ = Len(Str(c@))

    If d@ > e@ Then
      e@ = d@
      Print Using "____#"; b@; Using "___________#"; c@
    EndIf 
  Next
Return
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 upto 35500
Index-------Value
   37          11
 1173         108
35499        1076

0 OK, 0:145

Vala[edit]

Translation of: Nim
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);
    } 
  }
}
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[edit]

Translation of: C#
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
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[edit]

Library: Wren-fmt
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
}
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

XPL0[edit]

func IntLen(N); \Return number of digits in N
int  N, L;
[L:= 0;
repeat  N:= N/10;
        L:= L+1;
until   N = 0;
return L;
];

def Size = 1000000;
int Fusc(Size), N, Len, Max;
[Fusc(0):= 0;  Fusc(1):= 1;
for N:= 2 to Size-1 do
    Fusc(N):= if N&1 then Fusc((N-1)/2) + Fusc((N+1)/2) else Fusc(N/2);
for N:= 0 to 60 do
        [IntOut(0, Fusc(N));  ChOut(0, ^ )];
Text(0, "
n       fusc(n)
");
Max:= 0;
for N:= 0 to Size-1 do
    [Len:= IntLen(Fusc(N));
    if Len > Max then
        [Max:= Len;
        IntOut(0, N);  ChOut(0, 9\tab\);
        IntOut(0, Fusc(N));  CrLf(0);
        ];
    ];
]
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 
n       fusc(n)
0       0
37      11
1173    108
35499   1076
699051  10946

Yabasic[edit]

maximo = 20000000
dim f(maximo)

fusc()

for i = 0 to 60
    print f(i), " ";
next i

print "\n\n      Index       Value"
d = 0
for i = 0 to maximo-1
    if f(i) >= d then
        print i using "###,###,###", f(i) using "###,###,###"
        if d = 0 d = 1
        d = d * 10
    end if
next i
end

sub fusc()
    f(0) = 0 : f(1) = 1
    for n = 2 to maximo-1
        if mod(n, 2) then
            f(n) = f((n-1) / 2) + f((n+1) / 2)
        else
            f(n) = f(n / 2)
        end if
    next n
end sub
Output:
Igual que la entrada de FreeBASIC.

zkl[edit]

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 }
}
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