Counting integer points inside a sphere of radius R and dimension D

Question

I am trying to write an efficient algorithm that counts the number of points inside a Sphere of Radius R and Dimension D. The sphere is always at the origin. Suppose we have

Answer 1

An approach similar to that described by MBo, including source code, can be found at https://monsiterdex.wordpress.com/2013/04/05/integer-lattice-in-n-dimensional-sphere-count-of-points-with-integer-coordinates-using-parallel-programming-part-i/.

The approach consists in finding partitions of the radius, and then for each partition in the sphere compute the number of ways it can be represented in the sphere by both permuting coordinates and flipping the signs of nonzero coordinates.