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%I #22 Sep 19 2018 04:05:45
%S 0,0,2,8,24,68,188,516,1412,3860,10548,28820,78740,215124,587732,
%T 1605716,4386900,11985236,32744276,89459028,244406612,667731284,
%U 1824275796,4984014164,13616579924,37201188180,101635536212,277673448788,758617970004,2072582837588
%N Expansion of 2*x^2*(x+1) / (2*x^3-3*x+1).
%C Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n} that have annihilator elements.
%H Robert Israel, <a href="/A293006/b293006.txt">Table of n, a(n) for n = 0..2289</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_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-2).
%F a(n) = 2*A293005(n-1), a(0) = 0.
%F From _Colin Barker_, Sep 28 2017: (Start)
%F a(n) = (-8 + (1-sqrt(3))^(1+n) + (1+sqrt(3))^(1+n)) / 6 for n>0.
%F a(n) = 3*a(n-1) - 2*a(n-2) for n>3.
%F (End)
%p 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):
%p map(f, [$0..100]); # _Robert Israel_, Sep 28 2017
%t Join[{0}, LinearRecurrence[{3, 0, -2}, {0, 2, 8}, 30]] (* _Jean-François Alcover_, Sep 19 2018 *)
%o (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
%Y Cf. A002605, A028860, A293005, A293007.
%K nonn,easy
%O 0,3
%A _J. Devillet_, Sep 28 2017
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