Integer grid points intersection with sphere

Question

I'm wondering how many intersections does a centered sphere with radius $r$ ($r$ is an integer) have with an integer grid? For sure the 6 intersections with the axises, e.g. $(x,y,z)=(r,0,0)$. Actually, it should be enough to know how many intersections there are in the first quadrant, since the problem is symmetric.

Answer 1

What you are after is the set of Waterman polyhedra. Those are defined by the convex hull of the intersection of some lattice (you are interested in the simple cubic one) and a ball of increasing radius, centered at some lattice point.

For further info cf. to the corresponding wiki page: Waterman polyhedra
or esp. the relevant one on the website of Steve Waterman directly: simple cubic

--- rk