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 A104324 The Fibonacci word over the nonnegative integers; or, the number of runs of identical bits in the binary Zeckendorf representation of n. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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. 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, W. F. Smyth, More properties of the Fibonacci word on an infinite alphabet, arXiv:1710.02782 [math.CO], 2017. Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192. Jiemeng Zhang, Zhixiong Wen, Wen Wu, Some Properties of the Fibonacci Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics, 24(2) (2017), #P2.52. FORMULA a(n) = A007895(n) + A213911(n). - Reinhard Zumkeller, Mar 10 2013 EXAMPLE 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 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 *) PROG (Haskell) import Data.List (group) a104324 = length . map length . group . a213676_row -- Reinhard Zumkeller, Mar 10 2013 (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 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. Sequence in context: A237715 A238458 A182744 * A193212 A131818 A222111 Adjacent sequences:  A104321 A104322 A104323 * A104325 A104326 A104327 KEYWORD nonn,look AUTHOR Ron Knott, Mar 01 2005 EXTENSIONS Entry revised by N. J. A. Sloane, Jun 30 2017 STATUS approved

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Last modified August 16 16:40 EDT 2018. Contains 313809 sequences. (Running on oeis4.)