OFFSET
0,3
COMMENTS
Sequence has some interesting fractal properties (plot it!)
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..10000
Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
EXAMPLE
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.
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);
MATHEMATICA
Length @ Split[IntegerDigits[#, 2]] & /@ Select[Range[0, 1000], SequenceCount[ IntegerDigits[#, 2], {0, 0}] == 0 &] (* Amiram Eldar, Jan 18 2020 *)
CROSSREFS
AUTHOR
Ron Knott, Mar 01 2005
EXTENSIONS
Offset changed to 0 and a(0) prepended by Amiram Eldar, Jan 18 2020
STATUS
approved