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A286447 Number of inequivalent n X n matrices over GF(3) under action of dihedral group of the square D_4, with a third of 1's, 2's and 3's (ordered occurrences rounded up/down if n^2 != 0 mod 3). 8

%I #33 Apr 29 2019 05:18:53

%S 1,1,2,228,252642,3286762710,423091508279496,488322998306377824150,

%T 5405955851967092442258037800,561273297862912365721571649672300480,

%U 524055990531978935668322776302483856990581000

%N Number of inequivalent n X n matrices over GF(3) under action of dihedral group of the square D_4, with a third of 1's, 2's and 3's (ordered occurrences rounded up/down if n^2 != 0 mod 3).

%H María Merino, <a href="/A286447/b286447.txt">Table of n, a(n) for n = 0..45</a>

%H M. Merino and I. Unanue, <a href="https://doi.org/10.1387/ekaia.17851">Counting squared grid patterns with Pólya Theory</a>, EKAIA, 34 (2018), 289-316 (in Basque).

%F G.f.: g(x1,x2,x3) = 1/8*(y1^(n^2) + 2*y1^n*y2^((n^2 - n)/2) + 3*y2^(n^2/2) + 2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2) + 4*y1^n*y2^((n^2 - n)/2) + y1*y2^((n^2 - 1)/2) + 2*y1*y4^((n^2 - 1)/4)) if n odd, where coefficient correspond to y1 = x1 + x2 + x3, y2 = x1^2 + x2^2 + x3^2, y4 = x1^4 + x2^4 + x3^4 and occurrences of numbers are ceiling(n^2/3) for 1's and floor(n^2/3) for 2's and 3's.

%e For n=3 the a(3)=228 solutions are colorings of 3 X 3 matrices in 3 colors inequivalent under the action of D_4 with exactly 3 occurrences of each color (coefficient of x1^3 x2^3 x3^3).

%Y Cf. A054739, A082963.

%K nonn

%O 0,3

%A _María Merino_, Imanol Unanue, May 11 2017

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)