A319396 Number of partitions of n into exactly three positive Fibonacci numbers.

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, 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, 2, 2, 2, 3, 0, 2, 0, 0, 3, 1, 1, 1, 0, 3, 0, 0, 1, 0, 0

Offset: 0

Author

Alois P. Heinz, Sep 18 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..17711

Crossrefs

Column k=3 of A319394.

Cf. A000045.

Programs

Maple

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(3):
seq(a(n), n=0..120);

Mathematica

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]]];
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]]];
a[n_] := With[{k = 3}, b[n, h[n], k] - b[n, h[n], k - 1]];
a /@ Range[0, 120] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^3] 1/Product_{j>=2} (1-y*x^A000045(j)).