A129761 First differences of Fibbinary numbers (A003714).

1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 11, 1, 1, 2, 1, 3, 1, 1, 22, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 43, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 11, 1, 1, 2, 1, 3, 1, 1, 86, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 11, 1, 1, 2, 1, 3, 1, 1, 22, 1, 1, 2, 1, 3, 1, 1, 6, 1, 1, 2, 1, 171, 1, 1, 2, 1, 3, 1, 1, 6

Offset: 0

Author

Ralf Stephan, May 14 2007

Keywords

• nonn

Comments

Theorem: If the Zeckendorf representation of M ends with exactly k >= 0 zeros, ...10^k, then a(n) = ceiling(2^k/3). Also, if the Zeckendorf representation of n (A014417(n)) is even then a(n) is given by A319952, otherwise a(n) = 1. - Jeffrey Shallit and N. J. A. Sloane, Oct 03 2018

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (First 2500 terms from Vincenzo Librandi)

Crossrefs

Cf. A000045, A000071, A003714, A005578, A035614, A319432.

Programs

Maple

with(combinat): F:=fibonacci:
A072649:= proc(n) local j; global F; for j from ilog[(1+sqrt(5))/2](n)
       while F(j+1)<=n do od; (j-1); end:
A003714 := proc(n) global F; option remember; if(n < 3) then RETURN(n); else RETURN((2^(A072649(n)-1))+A003714(n-F(1+A072649(n)))); fi; end:
A129761 := n -> A003714(n+1)-A003714(n):
[seq(A129761(n),n=0..120)]; # N. J. A. Sloane, Oct 03 2018, borrowing programs from other sequences

Mathematica

Differences[Select[Range[600], !MemberQ[Partition[IntegerDigits[#, 2], 2, 1], {1, 1}] &]] (* Harvey P. Dale, Jul 17 2011 *)

Formula

a(n) = A005578(A035614(n)). - Alan Michael Gómez Calderón, Nov 01 2023

Extensions

a(0)=1 added by N. J. A. Sloane, Oct 02 2018