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 A257919 The number of combinatorial equivalence classes of n-endomorphisms on a rank-3 semigroup. 0

%I

%S 7,304,9958,288280,7973053,217032088,5875893676,158794026640,

%T 4288744989139,115807878426592,3126918614998354,84427755760664680,

%U 2279557984193621065,61548142781949118216,1661800549993751359192,44868621103769828836000,1211452826087259054393631

%N The number of combinatorial equivalence classes of n-endomorphisms on a rank-3 semigroup.

%C An n-endomorphism of a free semigroup is an endomorphism that sends every generator to a word of length <= n. Two n-endomorphisms are combinatorially equivalent if they are conjugate under an automorphism of the semigroup. This sequence gives the number of combinatorial equivalence classes of n-endomorphisms on a rank-3 semigroup, for n=1,2,3,...

%H Louis Rubin and Brian Rushton, <a href="http://arxiv.org/abs/1412.3001">Combinatorial Equivalence of m-Endomorphisms</a>, arXiv:1412.3001 [math.CO], 2014-2015.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (44,-553,2760,-6219,6156,-2187).

%F a(n) = (1/6)*(((3^(n+1)-3)/2)^3+3*n*((3^(n+1)-3)/2)+2*((3^(n+1)-3)/2)) = (3^n-1)*(12*n + 17 + 9*(9^n - 2*3^n))/16. [simplified by_Giovanni Resta_]

%F G.f.: x*(7 - 4*x + 453*x^2 - 1080*x^3)/((1 - 36*x + 243*x^2)*(1 - 4*x + 3*x^2)^2). [_Bruno Berselli_, May 19 2015]

%t Table[(3^n-1)(12 n + 17 + 9 (9^n - 2 3^n))/16, {n, 20}] (* _Giovanni Resta_, May 19 2015 *)

%o (MAGMA) [(3^n-1)*(12*n+17+9*(9^n-2*3^n))/16: n in [1..20]]; // _Bruno Berselli_, May 19 2015

%Y Cf. A134057, which gives the number of classes for a rank-2 semigroup.

%Y Cf. A006516, which gives the number of classes for a rank-2 monoid.

%K nonn,easy

%O 1,1

%A _Louis J. Rubin_, May 18 2015

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Last modified January 22 13:41 EST 2020. Contains 331149 sequences. (Running on oeis4.)