A241863 Number of compositions of n such that the smallest part has multiplicity three.

1, 0, 4, 5, 14, 24, 59, 108, 213, 419, 808, 1522, 2872, 5366, 9960, 18362, 33660, 61364, 111375, 201273, 362225, 649413, 1160289, 2066355, 3668840, 6495542, 11469453, 20201295, 35496670, 62233609, 108878818, 190103797, 331292391, 576296824, 1000766991

Offset: 3

Author

Joerg Arndt and Alois P. Heinz, Apr 30 2014

Keywords

• nonn

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A238342.

Cf. A001622.

Programs

Maple

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=3..40);

Mathematica

b[n_, s_] := b[n, s] = If[n == 0, 1, If[nJean-François Alcover, Feb 09 2015, after Maple *)

Formula

a(n) ~ (13*sqrt(5)-29)/300 * n^3 * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, May 01 2014

Equivalently, a(n) ~ n^3 * phi^(n-7) / 150, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021