A131661 Number of compositions of n such that the cardinality of the set of parts is 2.

0, 0, 2, 5, 14, 22, 44, 68, 107, 172, 261, 396, 606, 950, 1414, 2238, 3418, 5411, 8368, 13297, 20840, 33268, 52549, 84120, 133775, 214611, 343025, 551064, 883600, 1421767, 2284870, 3680296, 5924725, 9551161, 15393855, 24834827, 40061700

Offset: 1

Author

Vladeta Jovovic, Sep 13 2007

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A002133, A005772, A088142.

Column k=2 of A235998.

Cf. A242900 (with distinct multiplicities).

Programs

Maple

with(numtheory):
a:= n-> add(add(add(binomial(j+(n-i*j)/d, j), d=select(x->xAlois P. Heinz, Feb 01 2014

Mathematica

Rest@ CoefficientList[ Series[ Sum[ x^(i + j)*(x^i + x^j - 2)/((x^i - 1)*(x^j - 1)*(x^i + x^j - 1)), {i, 2, 37}, {j, i - 1}], {x, 0, 37}], x] (* Robert G. Wilson v, Sep 16 2007 *)

Formula

G.f.: Sum(Sum(x^(i+j)*(x^i+x^j-2)/((x^i-1)*(x^j-1)*(x^i+x^j-1)), j=1..i-1), i=2..infinity).

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

Extensions

More terms from Robert G. Wilson v, Sep 16 2007