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%I #57 Nov 04 2023 21:52:39
%S 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,
%T 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,
%U 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
%N First differences of Fibbinary numbers (A003714).
%C 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
%H N. J. A. Sloane, <a href="/A129761/b129761.txt">Table of n, a(n) for n = 0..10000</a> (First 2500 terms from Vincenzo Librandi)
%F a(n) = A005578(A035614(n)). - _Alan Michael Gómez Calderón_, Nov 01 2023
%p with(combinat): F:=fibonacci:
%p A072649:= proc(n) local j; global F; for j from ilog[(1+sqrt(5))/2](n)
%p while F(j+1)<=n do od; (j-1); end:
%p 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:
%p A129761 := n -> A003714(n+1)-A003714(n):
%p [seq(A129761(n),n=0..120)]; # _N. J. A. Sloane_, Oct 03 2018, borrowing programs from other sequences
%t Differences[Select[Range[600], !MemberQ[Partition[IntegerDigits[#, 2], 2, 1], {1, 1}] &]] (* _Harvey P. Dale_, Jul 17 2011 *)
%Y Cf. A000045, A000071, A003714, A005578, A035614, A319432.
%K nonn
%O 0,3
%A _Ralf Stephan_, May 14 2007
%E a(0)=1 added by _N. J. A. Sloane_, Oct 02 2018
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