A161026 Number of partitions of n into Fibonacci numbers where every part appears at least 2 times.

1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 8, 6, 12, 10, 15, 16, 22, 20, 30, 30, 38, 40, 51, 51, 67, 69, 84, 88, 108, 111, 136, 141, 168, 176, 209, 218, 256, 267, 310, 328, 376, 396, 454, 477, 541, 575, 644, 681, 767, 809, 902, 959, 1061, 1121, 1246, 1316, 1448, 1537, 1687, 1781, 1956

Offset

0,5

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(1+x^(2*F(j))/(1-x^(F(j))), j=2..infinity), where F = A000045 are the Fibonacci numbers. - Emeric Deutsch, Jun 24 2009

Example

a(9) = 5 because we have 333, 33111, 222111, 2211111, and 1^9. - Emeric Deutsch, Jun 24 2009

Maple

with(combinat); g := product(1+x^(2*fibonacci(j))/(1-x^fibonacci(j)), j = 2 .. 10): gser := series(g, x = 0, 95): seq(coeff(gser, x, n), n = 0 .. 65); # Emeric Deutsch, Jun 24 2009
# second Maple program:
F:= proc(n) option remember; (<<0|1>, <1|1>>^n)[1, 2] end:
b:= proc(n, i) option remember; `if`(n=0, 1, (f-> `if`(2*f<=n,
      add(b(n-j*f, i+1), j=[0, $2..n/f]), 0))(F(i)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 23 2019

Mathematica

F[n_] := F[n] = MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
b[n_, i_] := b[n, i] = If[n == 0, 1, With[{f = F[i]}, If[2*f <= n,
     Sum[b[n - j*f, i + 1], {j, Join[{0}, Range[2, n/f]]}], 0]]];
a[n_] := b[n, 2];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 26 2022, after Alois P. Heinz *)

Crossrefs

Cf. A000045.

Sequence in context: A128861 A161307 A161241 * A161077 A339220 A161293

Adjacent sequences: A161023 A161024 A161025 * A161027 A161028 A161029

Keyword

nonn

Author

R. H. Hardin, Jun 02 2009

Extensions

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

Status

approved