colorings of n^3 cubes

[Joshua Zucker]

Hi folks,
I'm wondering a few things, before I submit this sequence:
Is the description I've written here clear enough?
Is this sequence interesting enough to be submission-worthy?
What are good techniques for counting the number of nonnegative
solutions to a system of linear diophantine equations like this?  I've
just written a program to search the space by brute force, actually
listing every single solution as it counts.  If you have any
references to better algorithms I'd love to learn!

Thanks,
--Joshua Zucker


%I A000001
%S A000001 1,1,1,289,16743,2455061,9069797,25617192,60158319,122758692,227734800,398364425,
664106324,1063255428,1644644641,2469420200,3612955276,5166901813
%N A000001 Number of colorings of n^3 unit cubes in n colors such that
they can be assembled into a monochromatic nxnxn cube of any of the n
colors
%C A000001 Classify the n^3 cubes according to how many sides are
colored with each color.  In general, we can have 3-3, 3-2-1, 3-1-1-1,
2-2-2, 2-2-1-1, 2-1-1-1-1, and 1-1-1-1-1-1 cubes.  The set of cubes is
considered to be colored in the same way if the number of cubes in
each of these seven categories is the same.

Thus, also number of solutions of
2A + B + C = 8n (corners)
B + 3D + 2E + F = 4n(n-2) (edges)
B + 3C + 2E + 4F + 6G = 6(n-2)^2 (face interiors)
A+B+C+D+E+F+G = n^3 (total cubes; but this is redundant with the
previous equations, since every side of every cube must be painted)
%H A000001 Denise at <a
href="http://letsplaymath.wordpress.com/2008/03/19/leonhards-block-puzzles/">Let's
Play Math</a> blog asks how to color in the n=2 and n=3 cases
%e A000001 a(2) = 1 because all 8 blocks must be colored with 3 of one
color and 3 of another, so there's only 1 way to color the cubes.
%O A000001 1
%K A000001 ,nonn,
%A A000001 Joshua Zucker (joshua.zucker at stanfordalumni.org), May 22 2008