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A241863 Number of compositions of n such that the smallest part has multiplicity three. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 7 02:40 EST 2023. Contains 360111 sequences. (Running on oeis4.)