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

1, 2, 2, 4, 6, 14, 36, 114, 450, 2268, 14442, 118686, 1264678, 17519842, 318273566, 7607402556, 240151303078, 10055927801538, 559859566727028, 41582482495661986, 4129785050606801246, 549628445573614296188, 98256218721544814784486, 23631541930531250077261282

Offset: 0

Author

Paul D. Hanna, Oct 02 2007

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..125

Crossrefs

Cf. A000123, A000045.

Programs

Maple

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

Mathematica

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]]]];
f[n_] := With[{t = Floor@Log[2, 2n + 1]}, g[n/2^(t - 1), t]];
a[n_] := f[Fibonacci[n]];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 19 2020, after Alois P. Heinz *)

Formula

a(n) = A000123( A000045(n) ) for n>=0.