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A293005 Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}. 4

%I

%S 0,1,4,12,34,94,258,706,1930,5274,14410,39370,107562,293866,802858,

%T 2193450,5992618,16372138,44729514,122203306,333865642,912137898,

%U 2492007082,6808289962,18600594090,50817768106,138836724394,379308985002,1036291418794,2831200807594

%N Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}.

%H Colin Barker, <a href="/A293005/b293005.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 Jimmy Devillet, <a href="https://arxiv.org/abs/1712.07856">Bisymmetric and quasitrivial operations: characterizations and enumerations</a>, arXiv:1712.07856 [math.RA], 2017.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-2).

%F G.f.: x(x+1) / (2x^3-3*x+1).

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

%F 3*a(n)+2 = Sum_{k>=0} (2*binomial(n,2*k)+3*binomial(n,2*k+1))*3^k.

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

%F a(n) = (-4 - (1-sqrt(3))^n*(-2+sqrt(3)) + (1+sqrt(3))^n*(2+sqrt(3))) / 6.

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

%F (End)

%o (PARI) concat(0, Vec(x*(1 + x) / ((1 - x)*(1 - 2*x - 2*x^2)) + O(x^30))) \\ _Colin Barker_, Sep 28 2017

%Y Cf. A002605, A028860, A293006, A293007.

%K nonn,easy

%O 0,3

%A _J. Devillet_, Sep 28 2017

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Last modified April 21 07:53 EDT 2019. Contains 322327 sequences. (Running on oeis4.)