A289658 Number of 2-Abelian equivalence classes of words of length n over an alphabet of size 3.

1, 3, 9, 27, 75, 186, 414, 840, 1578, 2784, 4662, 7476, 11556, 17313, 25245, 35955, 50157, 68697, 92559, 122889, 161001, 208404, 266808, 338154, 424620, 528654, 652980, 800634, 974970, 1179699, 1418895, 1697037, 2019015, 2390175, 2816325, 3303783, 3859383, 4490526, 5205186, 6011964, 6920094

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Julien Cassaigne, J. Karhumäki, S. Puzynina, M. A. Whiteland, k-Abelian Equivalence and Rationality, in International Conference on Developments in Language Theory DLT 2016: Developments in Language Theory, pp. 77-88, Lecture Notes in Computer Science book series (LNCS, volume 9840).

Julien Cassaigne, Juhani Karhumaki, Svetlana Puzynina, Markus A. Whiteland, k-Abelian Equivalence and Rationality, Fundamenta Informaticae 154 (2017) 1-30, DOI 10.3233/FI-2017-1531.

Juhani Karhumäki, Markus A. Whiteland, Regularity of k-Abelian Equivalence Classes of Fixed Cardinality, Adventures Between Lower Bounds and Higher Altitudes, Lecture Notes in Computer Science, Vol. 11011, Springer, Cham, 49-62.

Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

Formula

See Maple code.

From Colin Barker, Jul 23 2017: (Start)

G.f.: (1 - 2*x + 2*x^2 + 6*x^3 - 2*x^4 - x^5 - 2*x^7 - x^8 + 2*x^9) / ((1 - x)^7*(1 + x)^2).

a(n) = (2*n^6 + 42*n^5 + 335*n^4 - 1140*n^3 + 5828*n^2 - 4752*n + 5760) / 1920 for n>0 and even.

a(n) = (2*n^6 + 42*n^5 + 335*n^4 - 1140*n^3 + 5828*n^2 - 4662*n + 5355) / 1920 for n odd.

(End)

Maple

# For n >= 1:
P23:=proc(n) local t1;
t1:=n^6/960+7*n^5/320+67*n^4/384-19*n^3/32+1457*n^2/480
-(1569/640+3*(-1)^n/128)*n+741/256+27*(-1)^n/256;
simplify(t1);
end;
[seq(P23(n), n=1..46)];

Mathematica

CoefficientList[Series[(1 - 2 x + 2 x^2 + 6 x^3 - 2 x^4 - x^5 - 2 x^7 - x^8 + 2 x^9)/((1 - x)^7*(1 + x)^2), {x, 0, 40}], x] (* Michael De Vlieger, Nov 29 2018 *)
LinearRecurrence[{5, -8, 0, 14, -14, 0, 8, -5, 1}, {1, 3, 9, 27, 75, 186, 414, 840, 1578, 2784}, 50] (* Harvey P. Dale, Aug 28 2019 *)

Prog

(PARI) Vec((1 - 2*x + 2*x^2 + 6*x^3 - 2*x^4 - x^5 - 2*x^7 - x^8 + 2*x^9) / ((1 - x)^7*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jul 23 2017

Crossrefs

Cf. A289657.

Sequence in context: A206604 A084707 A193703 * A180238 A289693 A269684

Adjacent sequences: A289655 A289656 A289657 * A289659 A289660 A289661

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 22 2017

Status

approved