A293005 Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}.
0, 1, 4, 12, 34, 94, 258, 706, 1930, 5274, 14410, 39370, 107562, 293866, 802858, 2193450, 5992618, 16372138, 44729514, 122203306, 333865642, 912137898, 2492007082, 6808289962, 18600594090, 50817768106, 138836724394, 379308985002, 1036291418794, 2831200807594
Offset: 0
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).
- Jimmy Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017.
- Index entries for linear recurrences with constant coefficients, signature (3,0,-2).
Programs
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PARI
concat(0, Vec(x*(1 + x) / ((1 - x)*(1 - 2*x - 2*x^2)) + O(x^30))) \\ Colin Barker, Sep 28 2017
Formula
G.f.: x(x+1) / (2x^3-3*x+1).
a(0) = 0, a(1) = 1, a(n+2)-2*a(n+1)-2*a(n) = 2.
3*a(n)+2 = Sum_{k>=0} (2*binomial(n,2*k)+3*binomial(n,2*k+1))*3^k.
From Colin Barker, Sep 28 2017: (Start)
a(n) = (-4 - (1-sqrt(3))^n*(-2+sqrt(3)) + (1+sqrt(3))^n*(2+sqrt(3))) / 6.
a(n) = 3*a(n-1) - 2*a(n-2) for n>2. (End)
E.g.f.: exp(x)*(2*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x) - 2)/3. - Stefano Spezia, Apr 11 2025