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%I #7 Dec 07 2020 16:03:26
%S 0,0,0,1,1,2,2,3,2,3,2,3,3,2,2,3,2,3,3,3,1,2,1,3,3,2,2,3,2,3,1,3,1,0,
%T 2,1,2,3,2,3,2,1,2,2,3,2,0,3,1,1,3,0,1,0,0,2,1,2,2,2,3,2,1,3,1,2,1,0,
%U 2,2,2,3,0,2,0,0,3,1,1,1,0,3,0,0,1,0,0
%N Number of partitions of n into exactly three positive Fibonacci numbers.
%H Alois P. Heinz, <a href="/A319396/b319396.txt">Table of n, a(n) for n = 0..17711</a>
%F a(n) = [x^n y^3] 1/Product_{j>=2} (1-y*x^A000045(j)).
%p h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
%p issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
%p end:
%p b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
%p t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
%p end:
%p a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(3):
%p seq(a(n), n=0..120);
%t 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 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 a[n_] := With[{k = 3}, b[n, h[n], k] - b[n, h[n], k - 1]];
%t a /@ Range[0, 120] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y Column k=3 of A319394.
%Y Cf. A000045.
%K nonn
%O 0,6
%A _Alois P. Heinz_, Sep 18 2018
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