A289657 Number of 3-abelian equivalence classes of words of length n over a binary alphabet.

1, 2, 4, 8, 16, 32, 60, 106, 176, 280, 426, 626, 892, 1238, 1678, 2230, 2910, 3738, 4734, 5920, 7318, 8954, 10852, 13040, 15546, 18400, 21632, 25276, 29364, 33932, 39016, 44654, 50884, 57748, 65286, 73542, 82560, 92386, 103066, 114650, 127186, 140726, 155322, 171028, 187898, 205990, 225360, 246068

Offset

0,2

Links

Ray Chandler, 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 (3, -2, -1, 0, 1, 2, -3, 1).

Formula

See Maple code.

G.f.: (1 - x + x^3 + 2*x^4 + 3*x^5 + x^7 - x^8 + 2*x^9) / ((1 - x)^5*(1 + x)*(1 + x + x^2)). - Colin Barker, Jul 23 2017

Maple

# This is for n >= 2:
P32:=proc(n) local t1, c, w; c:=exp(Pi*I/3); w:=c^2;
t1:=n^4/18-5*n^3/18+65*n^2/36-23*n/6-(-1)^n/8
+(2/27)*(w^n/c+c/w^n)+1307/216;
round(Re(evalf(simplify(t1))));
end;
[seq(P32(n), n=2..56)];

Mathematica

CoefficientList[Series[(1 - x + x^3 + 2 x^4 + 3 x^5 + x^7 - x^8 + 2 x^9)/((1 - x)^5*(1 + x) (1 + x + x^2)), {x, 0, 47}], x] (* Michael De Vlieger, Nov 29 2018 *)
LinearRecurrence[{3, -2, -1, 0, 1, 2, -3, 1}, {1, 2, 4, 8, 16, 32, 60, 106, 176, 280}, 50] (* Harvey P. Dale, Oct 06 2024 *)

Prog

(PARI) Vec((1 - x + x^3 + 2*x^4 + 3*x^5 + x^7 - x^8 + 2*x^9) / ((1 - x)^5*(1 + x)*(1 + x + x^2)) + O(x^50)) \\ Colin Barker, Jul 24 2017

Crossrefs

Cf. A289658.

Sequence in context: A196724 A056644 A007813 * A005309 A078389 A248847

Adjacent sequences: A289654 A289655 A289656 * A289658 A289659 A289660

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 22 2017

Status

approved