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Number of inequivalent n X n matrices over GF(3) under action of dihedral group of the square D_4.
9

%I #22 Apr 15 2021 15:27:19

%S 1,3,21,2862,5398083,105918450471,18761832172500795,

%T 29912416165371498901002,429210477536602279123636967061,

%U 55428311030379722725246681652572022523,64422190091501416379601522735200323789074174081,673878862467911703904942451533575765568815772023224550102

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

%H Andrew Howroyd, <a href="/A054739/b054739.txt">Table of n, a(n) for n = 0..25</a>

%F 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;

%F 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]

%t Join[{1, 3}, Table[CycleIndexPolynomial[

%t GraphData[{"Grid", {n, n}}, "AutomorphismGroup"],

%t Table[Subscript[s, i], {i, 1, 4}]] /.

%t Table[Subscript[s, i] -> 3, {i, 1, 4}], {n, 2, 10}]]

%t (* _Geoffrey Critzer_, Aug 09 2016 *)

%Y Column k=3 of A343097.

%Y Cf. A054247.

%K easy,nonn

%O 0,2

%A _Vladeta Jovovic_, May 15 2000

%E Terms a(10) and beyond from _Andrew Howroyd_, Apr 15 2021