Obtuse Angled Triangles

 Published on Friday, 26th September 2008, 10:00 pm; Solved by 1631;
Difficulty rating: 70%

Problem 210

Consider the set S(r) of points (x,y) with integer coordinates satisfying |x| + |y| ≤ r.
Let O be the point (0,0) and C the point (r/4,r/4).
Let N(r) be the number of points B in S(r), so that the triangle OBC has an obtuse angle, i.e. the largest angle α satisfies 90°<α<180°.
So, for example, N(4)=24 and N(8)=100.

What is N(1,000,000,000)?