I'm going to be thinking about these all day now.

This is akin to the day that I spent trying to come up with an analytic formula for the number of points on a regular grid within a circle of arbitrary radius centered on a gridpoint.

The best I could come up with was:

[2*floor(r/sqrt(2))+1]^2 + 4*[sum{n=floor(r/sqrt(2))+1 to n=floor(r) of floor(sqrt(r^2-n^2)) * 2 + 1}]

Prooving this formula is left as an exercise to the reader, as is coming up with a better one.