--description--

Consider the set $I_r$ of points $(x,y)$ with integer coordinates in the interior of the circle with radius $r$, centered at the origin, i.e. $x^2 + y^2 < r^2$.

For a radius of 2, $I_2$ contains the nine points (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1) and (1,-1). There are eight triangles having all three vertices in $I_2$ which contain the origin in the interior. Two of them are shown below, the others are obtained from these by rotation.

For a radius of 3, there are 360 triangles containing the origin in the interior and having all vertices in $I_3$ and for $I_5$ the number is 10600.

How many triangles are there containing the origin in the interior and having all three vertices in $I_{105}$?

--hints--

trianglesContainingOrigin() should return 1725323624056.

assert.strictEqual(trianglesContainingOrigin(), 1725323624056);

--seed--

--seed-contents--

function trianglesContainingOrigin() {

  return true;
}

trianglesContainingOrigin();

--solutions--

// solution required