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A319398 Number of partitions of n into exactly five positive Fibonacci numbers.

%I A319398 #7 Dec 08 2020 08:36:40
%S A319398 0,0,0,0,0,1,1,2,2,4,4,5,5,7,6,7,7,8,7,9,8,10,9,9,8,11,8,10,10,11,10,
%T A319398 11,10,13,10,11,8,10,10,10,11,12,11,11,11,13,11,12,11,12,12,11,11,13,
%U A319398 12,10,8,10,9,9,12,11,10,13,10,14,14,11,11,11,11,13
%N A319398 Number of partitions of n into exactly five positive Fibonacci numbers.
%H A319398 Alois P. Heinz, <a href="/A319398/b319398.txt">Table of n, a(n) for n = 0..17711</a>
%F A319398 a(n) = [x^n y^5] 1/Product_{j>=2} (1-y*x^A000045(j)).
%p A319398 h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
%p A319398       issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
%p A319398     end:
%p A319398 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
%p A319398       t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
%p A319398     end:
%p A319398 a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(5):
%p A319398 seq(a(n), n=0..120);
%t A319398 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 A319398 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 A319398 a[n_] := With[{k = 5}, b[n, h[n], k] - b[n, h[n], k - 1]];
%t A319398 a /@ Range[0, 120] (* _Jean-François Alcover_, Dec 08 2020 *)
%Y A319398 Column k=5 of A319394.
%Y A319398 Cf. A000045.
%K A319398 nonn
%O A319398 0,8
%A A319398 _Alois P. Heinz_, Sep 18 2018