OFFSET
1,1
COMMENTS
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,...
LINKS
Louis Rubin and Brian Rushton, Combinatorial Equivalence of m-Endomorphisms, arXiv:1412.3001 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (44,-553,2760,-6219,6156,-2187).
FORMULA
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_]
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]
MATHEMATICA
Table[(3^n-1)(12 n + 17 + 9 (9^n - 2 3^n))/16, {n, 20}] (* Giovanni Resta, May 19 2015 *)
PROG
(Magma) [(3^n-1)*(12*n+17+9*(9^n-2*3^n))/16: n in [1..20]]; // Bruno Berselli, May 19 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Louis J. Rubin, May 18 2015
STATUS
approved