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A104324 Number of runs (of equal bits) in the Zeckendorf (binary) representation of n. 7
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

1,2

COMMENTS

Series has some interesting fractal properties (plot it!)

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

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

REFERENCES

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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.

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.

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

CROSSREFS

Cf. A014417, A104325, A189920, A213676.

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

STATUS

approved

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Last modified June 27 11:25 EDT 2017. Contains 288788 sequences.