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A118113 Even Fibbinary numbers + 1; also 2*Fibbinary(n) + 1.

%I A118113 #29 Jun 14 2025 23:49:40
%S A118113 1,3,5,9,11,17,19,21,33,35,37,41,43,65,67,69,73,75,81,83,85,129,131,
%T A118113 133,137,139,145,147,149,161,163,165,169,171,257,259,261,265,267,273,
%U A118113 275,277,289,291,293,297,299,321,323,325,329,331,337,339,341,513,515,517
%N A118113 Even Fibbinary numbers + 1; also 2*Fibbinary(n) + 1.
%C A118113 m for which binomial(3*m-2,m) (see A117671) is odd, since by Kummer's theorem that happens exactly when the binary expansions of m and 2*m-2 have no 1 bit at the same position in each, and so m odd and no 11 bit pairs except optionally the least significant 2 bits. - _Kevin Ryde_, Jun 14 2025
%H A118113 Alois P. Heinz, <a href="/A118113/b118113.txt">Table of n, a(n) for n = 0..1000</a>
%F A118113 a(n) = A022340(n) + 1.
%F A118113 a(n) = 2*A003714(n) + 1.
%F A118113 Solutions to {x : binomial(3x,x) mod (x+1) != 0 } are given in A022341. The corresponding values of binomial(3x,x) mod (x+1) are given here.
%p A118113 F:= combinat[fibonacci]:
%p A118113 b:= proc(n) local j;
%p A118113       if n=0 then 0
%p A118113     else for j from 2 while F(j+1)<=n do od;
%p A118113          b(n-F(j))+2^(j-2)
%p A118113       fi
%p A118113     end:
%p A118113 a:= n-> 2*b(n)+1:
%p A118113 seq(a(n), n=0..70);  # _Alois P. Heinz_, Aug 03 2012
%t A118113 Select[Table[Mod[Binomial[3*k,k], k+1], {k,1200}], #>0&]
%Y A118113 Cf. A000108, A118112, A022341.
%Y A118113 Cf. A003714 (Fibbinary numbers), A022340 (even Fibbinary numbers).
%Y A118113 Cf. A117671, A263190, A171791, A263075.
%K A118113 nonn,easy
%O A118113 0,2
%A A118113 _Labos Elemer_, Apr 13 2006
%E A118113 New definition from _T. D. Noe_, Dec 19 2006