Search: id:A104325
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%I A104325
%S A104325 1,2,1,3,2,1,4,3,3,2,1,5,4,3,4,3,3,2,1,6,5,5,4,3,5,4,3,4,3,3,2,1,7,6,5,
%T A104325 6,5,5,4,3,6,5,5,4,3,5,4,3,4,3,3,2,1,8,7,7,6,5,7,6,5,6,5,5,4,3,7,6,5,6,
%U A104325 5,5,4,3,6,5,5,4,3,5,4,3,4,3,3,2,1,9,8,7,8,7,7,6,5,8,7,7,6,5
%N A104325 Number of runs of equal bits in the Dual Zeckendorf (binary) representation 
               of n.
%C A104325 Sequence has some interesting fractal properties (plot it!)
%H A104325 Ron Knott <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               fibrep.html">using Fibonacci Numbers to represent whole numbers</
               a>
%e A104325 The Dual Zeckendorf representation of 13 is 10110(fib) corresponding 
               to {8, 3, 2}
%e A104325 The largest set of Fibonacci numbers whose sum is n (cf. the Zeckendorf 
               rep is the smallest set). This is composed of runs of one 1, one 
               0, two 1's, one 0 i.e. 4 runs in all so a(13)=4
%p A104325 dualzeckrep:=proc(n)local i,z;z:=zeckrep(n);i:=1; while i<=nops(z)-2 
               do if z[i]=1 and z[i+1]=0 and z[i+2]=0 then z[i]:=0; z[i+1]:=1;z[i+2]:=1; 
               if i>3 then i:=i-2 fi else i:=i+1 fi od; if z[1]=0 then z:=subsop(1=NULL,
               z) fi; 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(dualzeckrep(n)),n=1..100);
%Y A104325 Cf. A014417, A104324.
%Y A104325 Sequence in context: A133334 A003603 A135227 this_sequence A133084 A118851 
               A112383
%Y A104325 Adjacent sequences: A104322 A104323 A104324 this_sequence A104326 A104327 
               A104328
%K A104325 nonn
%O A104325 1,2
%A A104325 Ron Knott (enquiry(AT)ronknott.com), Mar 01 2005

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