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

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 = 2..1000

Crossrefs

Column k=2 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=2..40);

Mathematica

b[n_, s_] := b[n, s] = If[n == 0, 1, If[nJean-François Alcover, Nov 07 2014, after Maple *)

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