

A104325


Number of runs of equal bits in the dual Zeckendorf representation of n (A104326).


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

Sequence has some interesting fractal properties (plot it!)


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 Fibonaccirelated sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175192.


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:=i2 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[i1] 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

Cf. A014417, A104324, A104326.
Sequence in context: A212536 A188277 A135227 * A204925 A133084 A118851
Adjacent sequences: A104322 A104323 A104324 * A104326 A104327 A104328


KEYWORD

nonn,hear,look


AUTHOR

Ron Knott, Mar 01 2005


EXTENSIONS

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jan 18 2020


STATUS

approved



