A104324 The Fibonacci word over the nonnegative integers; or, the number of runs of identical bits in the binary Zeckendorf representation of n.

0, 1, 2, 2, 3, 2, 3, 4, 2, 3, 4, 4, 5, 2, 3, 4, 4, 5, 4, 5, 6, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 6, 7, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 6, 7, 4, 5, 6, 6, 7, 6, 7, 8, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 6, 7, 4, 5, 6, 6, 7, 6, 7, 8, 4, 5, 6, 6, 7, 6, 7, 8, 6, 7, 8, 8, 9, 2, 3, 4, 4, 5, 4, 5, 6, 4, 5, 6, 6, 7, 4, 5, 6, 6

Offset: 0

Author

Ron Knott, Mar 01 2005

Keywords

• nonn
• look

Comments

Image of 0 under repeated application of the morphism phi = {2i -> 2i,2i+1; 2i+1 -> 2i+2: i = 0,1,2,3,...}. - N. J. A. Sloane, Jun 30 2017

This sequence has some interesting fractal properties (plot it!).

First occurrence of k=0,1,2,... is at 0,1,2,4,7,12,20,33,54, ..., A000071(k+1): Fibonacci numbers - 1. - Robert G. Wilson v, Apr 25 2006

Read mod 2 gives the Fibonacci word A003849. The differences, halved, give A213911.

Examples

			14 = 13+1 as a sum of Fibonacci numbers = 100001(in Fibonacci base) using the least number of 1's (Zeckendorf Rep): it consists of 3 runs: one 1, four 0's, one 1, so a(14)=3.
This sequence may be broken up into blocks of lengths 1,1,2,3,5,8,... (the nonzero Fibonacci numbers). The first occurrence of a number indicates the start of a new block. The first few blocks are:
0,
1,
2,2,
3,2,3,
4,2,3,4,4,
5,2,3,4,4,5,4,5,
6,2,3,4,4,5,4,5,6,4,5,6,6,
7,2,3,4,4,5,4,5,6,4,5,6,6,7,4,5,6,6,7,6,7,
8,2,3,4,4,5,4,5,6,4,5,6,6,7,4,5,6,6,7,6,7,8,4,5,6,6,7,6,7,8,6,7,8,8,
... (see also A288576). -  _N. J. A. Sloane_, Jun 30 2017
		

References

• E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..28656 [First 10000 terms from Reinhard Zumkeller]
• Amy Glen, Jamie Simpson, and W. F. Smyth, More properties of the Fibonacci word on an infinite alphabet, arXiv:1710.02782 [math.CO], 2017.
• Ron Knott, Using Fibonacci Numbers to Represent Whole Numbers
• Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
• Jiemeng Zhang, Zhixiong Wen, and Wen Wu, Some Properties of the Fibonacci Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics, 24(2) (2017), #P2.52.

Crossrefs

Cf. A007895, A014417, A104325, A189920, A213676, A213911.

See also the Fibonacci word A003849.

For partial sums see A288575.

See A288576 for another view of the initial blocks.

Programs

Haskell

import Data.List (group)
a104324 = length . map length . group . a213676_row
-- Reinhard Zumkeller, Mar 10 2013

Maple

with(combinat,fibonacci):fib:=fibonacci: zeckrep:=proc(N)local i,z,j,n;i:=2;z:=NULL;n:=N; while fib(i)<=n do i:=i+1 od;print(i=fib(i)); for j from i-1 by -1 to 2 do if n>=fib(j) then z:=z,1;n:=n-fib(j) else z:=z,0 fi od; [z] end proc: countruns:=proc(s)local i,c,elt;elt:=s[1];c:=1; for i from 2 to nops(s) do if s[i]<>s[i-1] then c:=c+1 fi od; c end proc: seq(countruns(zeckrep(n)),n=1..100);

Mathematica

f[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; While[ fr[[1]] == 0, fr = Rest@fr]; Length@ Split@ fr]; Array[f, 105] (* Robert G. Wilson v, Apr 25 2006 *)
Nest[ReplaceAll[#, {t_ /; EvenQ[t] :> Sequence[t, t+1], t_ /; OddQ[t] :> t+1}] &, {0}, 10] (* Paolo Xausa, Apr 05 2024 *)

PARI

phi(n) = if (n%2, n+1, [n, n+1]);
vphi(v) = nv = []; for (k=1, #v, nv = concat(nv, phi(v[k]));); nv;
lista(nn) = {v = [0]; for (i=1, nn, v = vphi(v);); v;} \\ Michel Marcus, Oct 10 2017

Formula

a(n) = A007895(n) + A213911(n). - Reinhard Zumkeller, Mar 10 2013

Extensions

Entry revised by N. J. A. Sloane, Jun 30 2017