random math-y thought: clearly, the first circle that hits any integer points at all is the one with radius 1, which hits the points (1,0), (-1,0), (0,1), and (0,-1). Four points total. A little later on, with radius sqrt(5), you hit eight points: (1,2), (2,1), (-1,2), (2,-1), (1,-2), (-2,1), (-1,-2), and (-2,-1)
What is the sequence of radii rn such that the circle at the origin with radius rn hits more integer points than any origin-centered circle with radius r < rn?
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hm, I suppose technically this should include the radius-zero circle as the first of the sequence
