Thanks for the input and for the explanation of Avni's work. I also thought my formulas were just estimates because of those fractions. However there is some logical justification in their creation. Perhaps an explanation will help and also may help justify Avni's solution if it is correct. My starting point is the idea that if a formula is possible, there must be a pattern in the results tied to N. Therefore there should also be a pattern in the error generated by (2^(n^2))/n^2. So I added an unknown X into the equation and solved for it using the data you supplied. The formula I used was: (2^(n^2) + X)/n^2 = a(n)
The result was: N X 1 0 2 12 3 64 4 960 5 768
I had already thought that even and odd boards needed different treatment since odd boards have a central square and even do not. The fact that n=4 yields a greater error than N=5 reinforced that belief. Then I noticed what I think is a significant pattern in the X values. In every case, when you break it down into smaller multiples, you end up with a power of 2 multiplied by N-1 and N+1. Example: 960 = (2^6)x3x5 This seems too coincidental to not be significant.