

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



1, 1, 2, 228, 252642, 3286762710, 423091508279496, 488322998306377824150, 5405955851967092442258037800, 561273297862912365721571649672300480, 524055990531978935668322776302483856990581000
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OFFSET

0,3


LINKS

María Merino, Table of n, a(n) for n = 0..45
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289316 (in Basque).


FORMULA

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.


EXAMPLE

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).


CROSSREFS

Cf. A054739, A082963.
Sequence in context: A071225 A212082 A015968 * A095220 A113800 A133495
Adjacent sequences: A286444 A286445 A286446 * A286448 A286449 A286450


KEYWORD

nonn


AUTHOR

María Merino, Imanol Unanue, May 11 2017


STATUS

approved



