A104325 Number of runs of equal bits in the dual Zeckendorf representation of n (A104326).
1, 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, 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, 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
Offset: 0
Examples
The dual Zeckendorf representation of 13 is 10110(fib) corresponding to {8, 3, 2}.
The largest set of Fibonacci numbers whose sum is n (cf. the Zeckendorf representation 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.
Links
- Amiram Eldar, Table of n, a(n) for n = 0..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.
Programs
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Maple
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);
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Mathematica
Length @ Split[IntegerDigits[#, 2]] & /@ Select[Range[0, 1000], SequenceCount[ IntegerDigits[#, 2], {0, 0}] == 0 &] (* Amiram Eldar, Jan 18 2020 *)
Extensions
Offset changed to 0 and a(0) prepended by Amiram Eldar, Jan 18 2020
Comments