A173285 A(x) satisfies: Fibonacci(x)/x = A(x)/A(x^2).

1, 1, 3, 4, 10, 14, 28, 42, 80, 122, 216, 338, 582, 920, 1544, 2464, 4088, 6552, 10762, 17314, 28292, 45606, 74236, 119842, 194660, 314502, 510082, 824584, 1336210, 2160794, 3499468, 5660262, 9163818, 14824080, 23994450, 38818530, 62823742, 101642272

Offset: 0

Author

Gary W. Adamson, Feb 14 2010

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000045, A173284.

Programs

Maple

A173285 := proc(n) option remember; if n = 0 then 1; else add(procname(l)*combinat[fibonacci](n-2*l+1),l=0..n/2) ; end if; end proc:
seq(A173285(n),n=0..60) ; # R. J. Mathar, Apr 01 2010

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[Fibonacci[n-2k+1] a[k], {k, 0, n/2}]];
a /@ Range[0, 40] (* Jean-François Alcover, Oct 02 2019 *)

Formula

a(n) = Sum_{k=0..n/2} A000045(n-2*k+1)*a(k). - R. J. Mathar, Apr 02 2010

Given M = triangle A173284, A173285 as a left-shifted vector = lim_{n->inf} M^n.

G.f.: Product_{k>=0} 1/(1 - x^(2^k) - x^(2^(k + 1))). - Ilya Gutkovskiy, Aug 30 2017

a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=1} 1/(1 - x^(2^k) - x^(2^(k+1))) = 2.6009165618094467356830434687244547021995030468423430186926... and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 08 2022

Extensions

Division through x added to definition and sequence extended by R. J. Mathar, Apr 22 2010