A359513 Number of partitions of n into at most 4 positive Fibonacci numbers (with a single type of 1).

1, 1, 2, 3, 4, 5, 6, 6, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 8, 8, 7, 8, 9, 7, 10, 8, 8, 9, 7, 8, 8, 4, 8, 5, 8, 9, 6, 10, 8, 6, 10, 6, 9, 8, 5, 9, 6, 6, 8, 4, 8, 4, 1, 8, 4, 7, 9, 5, 10, 7, 6, 10, 6, 8, 6, 3, 10, 5, 7, 9, 5, 8, 5, 2, 9, 4, 7, 6, 2, 8, 4, 3, 8, 1, 4, 1

Offset: 0

Author

Ilya Gutkovskiy, Jan 03 2023

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A000045, A003107, A319394, A319397, A359511, A359512, A359516.

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) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
a:= n-> (p-> add(coeff(p, x, i), i=0..4))(b(n, h(n))):
seq(a(n), n=0..87);  # Alois P. Heinz, Jan 03 2023

Mathematica

h[n_] := h[n] = If[n < 1, 0, With[{t = 5 n^2}, If[IntegerQ @ Sqrt[t + 4] || IntegerQ @ Sqrt[t - 4], n, h[n - 1]]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[x*b[n - i, h[Min[n - i, i]]]]];
a[n_] := Sum[Coefficient[#, x, i], {i, 0, 4}]&[b[n, h[n]]];
Table[a[n], {n, 0, 87}] (* Jean-François Alcover, May 26 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..4} A319394(n,k). - Alois P. Heinz, Jan 03 2023