A293006 Expansion of 2*x^2*(x+1) / (2*x^3-3*x+1).

0, 0, 2, 8, 24, 68, 188, 516, 1412, 3860, 10548, 28820, 78740, 215124, 587732, 1605716, 4386900, 11985236, 32744276, 89459028, 244406612, 667731284, 1824275796, 4984014164, 13616579924, 37201188180, 101635536212, 277673448788, 758617970004, 2072582837588

Offset: 0

Author

J. Devillet, Sep 28 2017

Keywords

• nonn
• easy

Comments

Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n} that have annihilator elements.

Links

• Robert Israel, Table of n, a(n) for n = 0..2289
• M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA] (2017).
• Index entries for linear recurrences with constant coefficients, signature (3,0,-2).

Crossrefs

Cf. A002605, A028860, A293005, A293007.

Programs

Maple

f:= gfun:-rectoproc({a(n) = 3*a(n-1) - 2*a(n-3),a(0)=0,a(1)=0,a(2)=2,a(3)=8},a(n),remember):
map(f, [$0..100]); # Robert Israel, Sep 28 2017

Mathematica

Join[{0}, LinearRecurrence[{3, 0, -2}, {0, 2, 8}, 30]] (* Jean-François Alcover, Sep 19 2018 *)

PARI

concat(vector(2), Vec(2*x^2*(1 + x) / ((1 - x)*(1 - 2*x - 2*x^2)) + O(x^30))) \\ Colin Barker, Sep 28 2017

Formula

a(n) = 2*A293005(n-1), a(0) = 0.

From Colin Barker, Sep 28 2017: (Start)

a(n) = (-8 + (1-sqrt(3))^(1+n) + (1+sqrt(3))^(1+n)) / 6 for n>0.

a(n) = 3*a(n-1) - 2*a(n-2) for n>3.

(End)