A238998 Number of partitions of n that such that no part is a Fibonacci number.

1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 2, 4, 4, 6, 5, 9, 8, 11, 11, 16, 16, 22, 22, 29, 31, 40, 42, 54, 57, 71, 77, 95, 103, 127, 137, 165, 182, 218, 238, 285, 313, 369, 408, 479, 530, 619, 684, 794, 883, 1019, 1130, 1304, 1446, 1658, 1843, 2107, 2340, 2670

Offset: 0

Author

Clark Kimberling, Mar 08 2014

Keywords

• nonn
• easy

Examples

			a(15) counts these partitions:  [15], [11,4], [9,6], [7,4,4]; a(16) counts these:  [16], [12,4], [10,6], [9,7], [6,6,4], [4,4,4,4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003107, A000045.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`((f-> issqr(f+4) or issqr(f-4))(5*d^2), 0, d),
        d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 31 2017

Mathematica

p[n_] := IntegerPartitions[n, All, Complement[Range@n, Fibonacci@Range@15]]; Table[p[n], {n, 0, 20}] (* shows partitions *)
a[n_] := Length@p@n; a /@ Range[0, 80] (* counts partitions *)
(* Second program: *)
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[
    If[Function[f, IntegerQ@Sqrt[f+4] || IntegerQ@Sqrt[f-4]][5*d^2], 0, d],
    {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
a /@ Range[0, 100] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz *)

PARI

N=66; q='q+O('q^N); Vec( prod(n=1,11,1-q^fibonacci(n+1))/eta(q) ) \\ Joerg Arndt, Mar 11 2014

Formula

G.f.: A(x) = sum(1/product(1 - x^c(i))), i >=1, where c(i) are the non-Fibonacci numbers.