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A238998 Number of partitions of n that such that no part is a Fibonacci number.

%I A238998 #19 Jun 05 2021 06:28:50
%S A238998 1,0,0,0,1,0,1,1,1,1,2,2,3,2,4,4,6,5,9,8,11,11,16,16,22,22,29,31,40,
%T A238998 42,54,57,71,77,95,103,127,137,165,182,218,238,285,313,369,408,479,
%U A238998 530,619,684,794,883,1019,1130,1304,1446,1658,1843,2107,2340,2670
%N A238998 Number of partitions of n that such that no part is a Fibonacci number.
%H A238998 Alois P. Heinz, <a href="/A238998/b238998.txt">Table of n, a(n) for n = 0..10000</a>
%F A238998 G.f.:  A(x) = sum(1/product(1 - x^c(i))), i >=1, where c(i) are the non-Fibonacci numbers.
%e A238998 a(15) counts these partitions:  [15], [11,4], [9,6], [7,4,4]; a(16) counts these:  [16], [12,4], [10,6], [9,7], [6,6,4], [4,4,4,4].
%p A238998 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A238998       `if`((f-> issqr(f+4) or issqr(f-4))(5*d^2), 0, d),
%p A238998         d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A238998     end:
%p A238998 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 31 2017
%t A238998 p[n_] := IntegerPartitions[n, All, Complement[Range@n, Fibonacci@Range@15]]; Table[p[n], {n, 0, 20}] (* shows partitions *)
%t A238998 a[n_] := Length@p@n; a /@ Range[0, 80] (* counts partitions *)
%t A238998 (* Second program: *)
%t A238998 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[
%t A238998     If[Function[f, IntegerQ@Sqrt[f+4] || IntegerQ@Sqrt[f-4]][5*d^2], 0, d],
%t A238998     {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
%t A238998 a /@ Range[0, 100] (* _Jean-François Alcover_, Jun 05 2021, after _Alois P. Heinz_ *)
%o A238998 (PARI) N=66; q='q+O('q^N); Vec( prod(n=1,11,1-q^fibonacci(n+1))/eta(q) ) \\ _Joerg Arndt_, Mar 11 2014
%Y A238998 Cf. A003107, A000045.
%K A238998 nonn,easy
%O A238998 0,11
%A A238998 _Clark Kimberling_, Mar 08 2014