I'm trying to solve a problem by using combinatorics. The problem goes like this:
I want to place $m$ balls in a $n\times n\times n$ chessboard. No two balls should be adjacent. Thus, no two of them should be the nearest neighbor (each location in the chessboard has six nearest neighbors).
For example: if a ball is placed at (1,1,1), there must be no ball at (0,1,1), (1,0,1), (1,2,1), (2,1,1), (1,1,0), (1,1,2).
For the ball at the corner location, like (0,0,0), there must be no ball at (0,0,1), (0,1,0) and (1,0,0).
How many different ways to put these $m$ balls? Is this problem solvable in general? Is it possible to write down a generating series for such problem?