--description--

For any triangle $T$ in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside $T$.

For a given $n$, consider triangles $T$ such that:

• the vertices of $T$ have integer coordinates with absolute value $≤ n$, and
• the foci¹ of the largest-area ellipse inside $T$ are $(\sqrt{13}, 0)$ and $(-\sqrt{13}, 0)$.

Let $A(n)$ be the sum of the areas of all such triangles.

For example, if $n = 8$, there are two such triangles. Their vertices are (-4,-3), (-4,3), (8,0) and (4,3), (4,-3), (-8,0), and the area of each triangle is 36. Thus $A(8) = 36 + 36 = 72$.

It can be verified that $A(10) = 252$, $A(100) = 34\,632$ and $A(1000) = 3\,529\,008$.

Find $A(1\,000\,000\,000)$.

¹The foci (plural of focus) of an ellipse are two points $A$ and $B$ such that for every point $P$ on the boundary of the ellipse, $AP + PB$ is constant.

--hints--

ellipsesInsideTriangles() should return 3776957309612154000.

assert.strictEqual(ellipsesInsideTriangles(), 3776957309612154000);

--seed--

--seed-contents--

function ellipsesInsideTriangles() {

  return true;
}

ellipsesInsideTriangles();

--solutions--

// solution required