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A293007 Expansion of 2*x^2 / (1-2*x-2*x^2). 4


%S 0,0,2,4,12,32,88,240,656,1792,4896,13376,36544,99840,272768,745216,

%T 2035968,5562368,15196672,41518080,113429504,309895168,846649344,

%U 2313089024,6319476736,17265131520,47169216512,128868696064,352075825152,961889042432

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

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

%H Colin Barker, <a href="/A293007/b293007.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Couceiro, J. Devillet, and J.-L. Marichal, <a href="http://arxiv.org/abs/1709.09162">Quasitrivial semigroups: characterizations and enumerations</a>, arXiv:1709.09162 [math.RA] (2017).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,2).

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

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

%F From _Colin Barker_, Sep 28 2017: (Start)

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

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

%F (End)

%o (PARI) concat(vector(2), Vec(2*x^2 / (1-2*x-2*x^2) + O(x^50))) \\ _Colin Barker_, Sep 28 2017

%Y Cf. A002605, A293005, A293006.

%Y Essentially the same as A028860 and A152035.

%K nonn,easy

%O 0,3

%A _J. Devillet_, Sep 28 2017

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Last modified August 8 11:12 EDT 2020. Contains 336293 sequences. (Running on oeis4.)