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

%I A173285 #18 Oct 08 2022 06:26:47
%S A173285 1,1,3,4,10,14,28,42,80,122,216,338,582,920,1544,2464,4088,6552,10762,
%T A173285 17314,28292,45606,74236,119842,194660,314502,510082,824584,1336210,
%U A173285 2160794,3499468,5660262,9163818,14824080,23994450,38818530,62823742,101642272
%N A173285 A(x) satisfies: Fibonacci(x)/x = A(x)/A(x^2).
%H A173285 Alois P. Heinz, <a href="/A173285/b173285.txt">Table of n, a(n) for n = 0..2000</a>
%F A173285 a(n) = Sum_{k=0..n/2} A000045(n-2*k+1)*a(k). - _R. J. Mathar_, Apr 02 2010
%F A173285 Given M = triangle A173284, A173285 as a left-shifted vector = lim_{n->inf} M^n.
%F A173285 G.f.: Product_{k>=0} 1/(1 - x^(2^k) - x^(2^(k + 1))). - _Ilya Gutkovskiy_, Aug 30 2017
%F A173285 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
%p A173285 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:
%p A173285 seq(A173285(n),n=0..60) ; # _R. J. Mathar_, Apr 01 2010
%t A173285 a[n_] := a[n] = If[n == 0, 1, Sum[Fibonacci[n-2k+1] a[k], {k, 0, n/2}]];
%t A173285 a /@ Range[0, 40] (* _Jean-François Alcover_, Oct 02 2019 *)
%Y A173285 Cf. A000045, A173284.
%K A173285 nonn
%O A173285 0,3
%A A173285 _Gary W. Adamson_, Feb 14 2010
%E A173285 Division through x added to definition and sequence extended by _R. J. Mathar_, Apr 22 2010