--description--

A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:

$$x(t) = (R - r) \cos(t) + r \cos(\frac{R - r}{r}t)$$

$$y(t) = (R - r) \sin(t) - r \sin(\frac{R - r}{r} t)$$

Where $R$ is the radius of the large circle and $r$ the radius of the small circle.

Let $C(R, r)$ be the set of distinct points with integer coordinates on the hypocycloid with radius $R$ and $r$ and for which there is a corresponding value of $t$ such that $\sin(t)$ and $\cos(t)$ are rational numbers.

Let $S(R, r) = \sum_{(x,y) \in C(R, r)} |x| + |y|$ be the sum of the absolute values of the $x$ and $y$ coordinates of the points in $C(R, r)$.

Let $T(N) = \sum_{R = 3}^N \sum_{r=1}^{\left\lfloor \frac{R - 1}{2} \right\rfloor} S(R, r)$ be the sum of $S(R, r)$ for $R$ and $r$ positive integers, $R\leq N$ and $2r < R$.

You are given:

$$\begin{align} C(3, 1) = & \{(3, 0), (-1, 2), (-1,0), (-1,-2)\} \\ C(2500, 1000) = & \{(2500, 0), (772, 2376), (772, -2376), (516, 1792), (516, -1792), (500, 0), (68, 504), \\ &(68, -504),(-1356, 1088), (-1356, -1088), (-1500, 1000), (-1500, -1000)\} \end{align}$$

Note: (-625, 0) is not an element of $C(2500, 1000)$ because $\sin(t)$ is not a rational number for the corresponding values of $t$.

$S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10$

$T(3) = 10$; $T(10) = 524$; $T(100) = 580\,442$; $T({10}^3) = 583\,108\,600$.

Find $T({10}^6)$.

--hints--

hypocycloidAndLatticePoints() should return 583333163984220900.

assert.strictEqual(hypocycloidAndLatticePoints(), 583333163984220900);

--seed--

--seed-contents--

function hypocycloidAndLatticePoints() {

  return true;
}

hypocycloidAndLatticePoints();

--solutions--

// solution required