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), 289-316 (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
KEYWORD
nonn
AUTHOR
María Merino, Imanol Unanue, May 11 2017
STATUS
approved