--description--

The triangle $ΔABC$ is inscribed in an ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $0 < 2b < a$, $a$ and $b$ integers.

Let $r(a, b)$ be the radius of the incircle of $ΔABC$ when the incircle has center $(2b, 0)$ and $A$ has coordinates $\left(\frac{a}{2}, \frac{\sqrt{3}}{2}b\right)$.

For example, $r(3, 1) = \frac{1}{2}, r(6, 2) = 1, r(12, 3) = 2$.

Let $G(n) = \sum_{a = 3}^n \sum_{b = 1}^{\left\lfloor\frac{a - 1}{2} \right\rfloor} r(a, b)$

You are given $G(10) = 20.59722222$, $G(100) = 19223.60980$ (rounded to 10 significant digits).

Find $G({10}^{11})$. Give your answer as a string in scientific notation rounded to 10 significant digits. Use a lowercase e to separate mantissa and exponent.

For $G(10)$ the answer would have been 2.059722222e1

--hints--

triangleInscribedInEllipse() should return a string.

assert.isString(triangleInscribedInEllipse());

triangleInscribedInEllipse() should return the string 1.895093981e31.

assert.strictEqual(triangleInscribedInEllipse(), '1.895093981e31');

--seed--

--seed-contents--

function triangleInscribedInEllipse() {

  return true;
}

triangleInscribedInEllipse();

--solutions--

// solution required