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A238998
Number of partitions of n that such that no part is a Fibonacci number.
2
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
0,11
LINKS
FORMULA
G.f.: A(x) = sum(1/product(1 - x^c(i))), i >=1, where c(i) are the non-Fibonacci numbers.
EXAMPLE
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].
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 *)
PROG
(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
CROSSREFS
Sequence in context: A117660 A358015 A184198 * A355359 A144369 A033822
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 08 2014
STATUS
approved