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A134041 a(n) = number of binary partitions of the Fibonacci number F(n).

%I A134041 #17 Nov 19 2020 11:56:30
%S A134041 1,2,2,4,6,14,36,114,450,2268,14442,118686,1264678,17519842,318273566,
%T A134041 7607402556,240151303078,10055927801538,559859566727028,
%U A134041 41582482495661986,4129785050606801246,549628445573614296188,98256218721544814784486,23631541930531250077261282
%N A134041 a(n) = number of binary partitions of the Fibonacci number F(n).
%H A134041 Alois P. Heinz, <a href="/A134041/b134041.txt">Table of n, a(n) for n = 0..125</a>
%F A134041 a(n) = A000123( A000045(n) ) for n>=0.
%p A134041 g:= proc(b, n) option remember; local t; if b<0 then 0 elif b=0 or n=0 then 1 elif b>=n then add(g(b-t, n) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1) else g(b-1, n) +g(2*b, n-1) fi end: f:= proc(n) local t; t:= ilog2(2*n+1); g(n/2^(t-1), t) end: a:= n-> f(combinat[fibonacci](n)): seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 26 2011
%t A134041 g[b_, n_] := g[b, n] = If[b < 0, 0, If[b == 0 || n == 0, 1, If[b >= n, Sum[g[b - t, n] Binomial[n + 1, t] (-1)^(t + 1), {t, 1, n + 1}], g[b - 1, n] + g[2b, n - 1]]]];
%t A134041 f[n_] := With[{t = Floor@Log[2, 2n + 1]}, g[n/2^(t - 1), t]];
%t A134041 a[n_] := f[Fibonacci[n]];
%t A134041 a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 19 2020, after _Alois P. Heinz_ *)
%Y A134041 Cf. A000123, A000045.
%K A134041 nonn
%O A134041 0,2
%A A134041 _Paul D. Hanna_, Oct 02 2007