A241862 Number of compositions of n such that the smallest part has multiplicity two.

1, 0, 4, 3, 10, 18, 35, 60, 121, 217, 391, 709, 1281, 2283, 4061, 7185, 12680, 22291, 39051, 68191, 118767, 206331, 357596, 618345, 1066983, 1837513, 3158685, 5420335, 9286086, 15884155, 27130404, 46274560, 78822938, 134095757, 227853408, 386721362, 655639944

Offset

2,3

Links

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 2..1000

Formula

a(n) ~ (1/4 - 11/(20*sqrt(5))) * n^2 * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, May 01 2014

Equivalently, a(n) ~ n^2 * phi^(n-5) / (2 * 5^(3/2)), 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:= 2;
      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=2..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=2}, 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, 2, 40}] (* Jean-François Alcover, Nov 07 2014, after Maple *)

Crossrefs

Column k=2 of A238342.

Cf. A001622.

Sequence in context: A185730 A205965 A242391 * A222510 A100492 A072183

Adjacent sequences: A241859 A241860 A241861 * A241863 A241864 A241865

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Apr 30 2014

Status

approved