Obtuse Angled Triangles
Consider the set $S(r)$ of points $(x,y)$ with integer coordinates satisfying $|x| + |y| \le r$.
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 $\alpha$ satisfies $90^\circ \lt \alpha \lt 180^\circ$.
So, for example, $N(4)=24$ and $N(8)=100$.
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 $\alpha$ satisfies $90^\circ \lt \alpha \lt 180^\circ$.
So, for example, $N(4)=24$ and $N(8)=100$.
What is $N(1\,000\,000\,000)$?