A054739 Number of inequivalent n X n matrices over GF(3) under action of dihedral group of the square D_4.

1, 3, 21, 2862, 5398083, 105918450471, 18761832172500795, 29912416165371498901002, 429210477536602279123636967061, 55428311030379722725246681652572022523, 64422190091501416379601522735200323789074174081, 673878862467911703904942451533575765568815772023224550102

Offset: 0

Author

Vladeta Jovovic, May 15 2000

Keywords

• easy
• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..25

Crossrefs

Column k=3 of A343097.

Cf. A054247.

Programs

Mathematica

Join[{1, 3}, Table[CycleIndexPolynomial[
    GraphData[{"Grid", {n, n}}, "AutomorphismGroup"],
    Table[Subscript[s, i], {i, 1, 4}]] /.
Table[Subscript[s, i] -> 3, {i, 1, 4}], {n, 2, 10}]]
(* Geoffrey Critzer, Aug 09 2016 *)

Formula

a(n) = (1/8)*(3^(n^2) + 2*3^(n^2/4) + 3*3^(n^2/2) + 2*3^((n^2+n)/2)) if n is even;

a(n) = (1/8)*(3^(n^2) + 2*3^((n^2+3)/4) + 3^((n^2+1)/2) + 4*3^((n^2+n)/2)) if n is odd. [corrected by Chris Hallstrom, Mar 22 2021]

Extensions

Terms a(10) and beyond from Andrew Howroyd, Apr 15 2021