A286525 Number of inequivalent n X n matrices over GF(4) under action of dihedral group of the square D_4, with a fourth of 1's, 2's, 3's and 4's (ordered occurrences rounded up/down if n^2 != 0 mod 4).

1, 1, 3, 978, 7885536, 1030690752000, 2681594035175055000, 111102459342780333711432912, 82765346051371433995689422809152600, 984929152509556378339959477248973638627262816, 201525938526971993585665495909682003042353826154218776128

Offset

0,3

Links

María Merino, Table of n, a(n) for n = 0..39

M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Formula

G.f.: g(x1,x2,x3,x4) = 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 + x4, y2 = x1^2 + x2^2 + x3^2 + x4^2, y4 = x1^4 + x2^4 + x3^4 + x4^2 and occurrences of numbers are ceiling(n^2/4) for 1's and floor(n^2/4) for 2's, 3's and 4's.

Example

For n=2 the a(2)=3 solutions are the colorings of 2 X 2 matrices in 4 colors inequivalent under the action of D_4 with exactly 1 occurrence of each color (coefficient of x1^1 x2^1 x3^1 x4^1).

Crossrefs

Cf. A054751, A082963, A286447.

Sequence in context: A167067 A324443 A151585 * A030250 A145231 A318480

Adjacent sequences: A286522 A286523 A286524 * A286526 A286527 A286528

Keyword

nonn

Author

María Merino, Imanol Unanue, May 11 2017

Status

approved