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A319395 Number of partitions of n into exactly two positive Fibonacci numbers.

%I A319395 #9 Dec 07 2020 14:52:43
%S A319395 0,0,1,1,2,1,2,1,1,1,2,1,0,1,1,1,2,0,1,0,0,1,1,1,1,0,2,0,0,1,0,0,0,0,
%T A319395 1,1,1,1,0,1,0,0,2,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,0,1,0,0,1,0,0,0,0,
%U A319395 2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0
%N A319395 Number of partitions of n into exactly two positive Fibonacci numbers.
%H A319395 Alois P. Heinz, <a href="/A319395/b319395.txt">Table of n, a(n) for n = 0..17711</a>
%F A319395 a(n) = [x^n y^2] 1/Product_{j>=2} (1-y*x^A000045(j)).
%p A319395 h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
%p A319395       issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
%p A319395     end:
%p A319395 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
%p A319395       t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
%p A319395     end:
%p A319395 a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(2):
%p A319395 seq(a(n), n=0..120);
%t A319395 h[n_] := h[n] = If[n < 1, 0, If[Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2], n, h[n - 1]]];
%t A319395 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
%t A319395 a[n_] := With[{k = 2}, b[n, h[n], k] - b[n, h[n], k - 1]];
%t A319395 a /@ Range[0, 120] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A319395 Column k=2 of A319394.
%Y A319395 Cf. A000045.
%K A319395 nonn
%O A319395 0,5
%A A319395 _Alois P. Heinz_, Sep 18 2018