A238998 - OEIS

%I #19 Jun 05 2021 06:28:50

%S 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,

%T 42,54,57,71,77,95,103,127,137,165,182,218,238,285,313,369,408,479,

%U 530,619,684,794,883,1019,1130,1304,1446,1658,1843,2107,2340,2670

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

%H Alois P. Heinz, <a href="/A238998/b238998.txt">Table of n, a(n) for n = 0..10000</a>

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

%e 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].

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p       `if`((f-> issqr(f+4) or issqr(f-4))(5*d^2), 0, d),

%p         d=numtheory[divisors](j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 31 2017

%t p[n_] := IntegerPartitions[n, All, Complement[Range@n, Fibonacci@Range@15]]; Table[p[n], {n, 0, 20}] (* shows partitions *)

%t a[n_] := Length@p@n; a /@ Range[0, 80] (* counts partitions *)

%t (* Second program: *)

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[

%t     If[Function[f, IntegerQ@Sqrt[f+4] || IntegerQ@Sqrt[f-4]][5*d^2], 0, d],

%t     {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];

%t a /@ Range[0, 100] (* _Jean-François Alcover_, Jun 05 2021, after _Alois P. Heinz_ *)

%o (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

%Y Cf. A003107, A000045.

%K nonn,easy

%O 0,11

%A _Clark Kimberling_, Mar 08 2014