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

3,3

Links

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

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

Maple

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
    end:
a:= proc(n) local k; k:= 3;
      add((p->add(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[n<s, 0, Expand[Sum[b[n-j, s]*x, {j, s, n}]]]]; a[n_] := With[{k=3}, Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i + k, k], {i, 0, Exponent[p, x]}]][b[n-j*k, j+1]], {j, 1, n/k}]]; Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

Crossrefs

Column k=3 of A238342.

Cf. A001622.

Sequence in context: A348889 A006904 A200177 * A350527 A316415 A007084

Adjacent sequences: A241860 A241861 A241862 * A241864 A241865 A241866

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Apr 30 2014

Status

approved