A293007 Expansion of 2*x^2 / (1-2*x-2*x^2).

0, 0, 2, 4, 12, 32, 88, 240, 656, 1792, 4896, 13376, 36544, 99840, 272768, 745216, 2035968, 5562368, 15196672, 41518080, 113429504, 309895168, 846649344, 2313089024, 6319476736, 17265131520, 47169216512, 128868696064, 352075825152, 961889042432

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 neutral and annihilator elements.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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 (2,2).

Crossrefs

Cf. A002605, A293005, A293006.

Essentially the same as A028860 and A152035.

Programs

PARI

concat(vector(2), Vec(2*x^2 / (1-2*x-2*x^2) + O(x^50))) \\ Colin Barker, Sep 28 2017

Formula

a(n) = 2*A002605(n-1), a(0) = 0.

a(n) = A028860(n+1), a(0) = 0.

From Colin Barker, Sep 28 2017: (Start)

a(n) = ((1-sqrt(3))^n*(1+sqrt(3)) + (-1+sqrt(3))*(1+sqrt(3))^n) / (2*sqrt(3)) for n>0.

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

(End)