Search: id:A104324
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%I A104324
%S A104324 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,
%T A104324 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,
%U A104324 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
%N A104324 Number of runs (of equal bits) in the Zeckendorf (binary) representation 
               of n.
%C A104324 Series has some interesting fractal properties (plot it!)
%C A104324 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.
%D A104324 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.
%H A104324 R. Knott <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               fibrep.html">Using Fibonacci Numbers to Represent Whole Numbers</
               a>
%e A104324 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
%p A104324 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);
%t A104324 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] (from Robert 
               G. Wilson v (rgwv(at)rgwv.com), Apr 25 2006)
%Y A104324 Cf. A014417, A104325.
%Y A104324 Sequence in context: A046773 A101037 A002199 this_sequence A131818 A070081 
               A034883
%Y A104324 Adjacent sequences: A104321 A104322 A104323 this_sequence A104325 A104326 
               A104327
%K A104324 nonn
%O A104324 1,2
%A A104324 Ron Knott (enquiry(AT)ronknott.com), Mar 01 2005

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