Hmm...no action on this one. Here's the solution:
First note that the three vectors I have given do span 3 space, even if they aren't orthogonal, so they do the trick for a basis. If you would like you can change the basis simply use
and use the usual orthonormal set of basis vectors, but where's the fun in that?
Now, I don't know about you but messing around with non-orthogonal basis systems is a little out of my league. (I couldn't make it work anyway...I never was good with the crystal groups.) But we can always go back to the stand-by:
and we know that
We get three sets of three equations and they're pretty easy to solve. I get
That's the east part. To find a pattern we need to do a bunch of calculations. If you do this on paper (c'mon! Feel the burn!) it takes a little while, but I get:
, . This gives a pattern for the odds: .
We will need the matrix :
We now find
, . This gives a pattern for the evens: