A161027 Number of partitions of n into Fibonacci numbers where every part appears at least 3 times.

1, 0, 0, 1, 1, 1, 2, 1, 2, 3, 3, 3, 6, 5, 6, 10, 8, 9, 14, 13, 16, 20, 19, 23, 30, 30, 33, 41, 43, 48, 59, 58, 67, 78, 81, 92, 105, 109, 123, 140, 148, 160, 182, 193, 214, 238, 249, 275, 305, 322, 353, 386, 413, 447, 490, 520, 561, 611, 650, 701, 762, 804, 868, 938, 997, 1067, 1147

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from R. H. Hardin)

Formula

G.f.: Product_{j>=2} (1 + x^(3*F(j))/(1 - x^(F(j)))), where F = A000045 are the Fibonacci numbers. - Emeric Deutsch, Jun 23 2009

Example

a(10) = 3 because we have 22222, 2221111, and 1^(10). - Emeric Deutsch, Jun 23 2009

Maple

with(combinat): g := product(1+x^(3*fibonacci(j))/(1-x^fibonacci(j)), j = 2 .. 10): gser := series(g, x = 0, 95): seq(coeff(gser, x, n), n = 0 .. 71); # Emeric Deutsch, Jun 23 2009
# second Maple program:
F:= proc(n, i) option remember; (<<0|1>, <1|1>>^n)[1, 2] end:
b:= proc(n, i) option remember; `if`(n=0, 1, (f-> `if`(3*f<=n,
      add(b(n-j*f, i+1), j=[0, $3..n/f]), 0))(F(i)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 23 2019

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, With[{f = Fibonacci[i]}, If[3*f <= n,       Sum[b[n - j*f, i + 1], {j, Prepend[Range[3, n/f], 0]}], 0]]];
a[n_] := b[n, 2];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 30 2023, after Alois P. Heinz *)

Crossrefs

Cf. A000045.

Sequence in context: A306878 A161308 A161242 * A161078 A161294 A161269

Adjacent sequences: A161024 A161025 A161026 * A161028 A161029 A161030

Keyword

nonn

Author

R. H. Hardin, Jun 02 2009

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 23 2019

Status

approved