Hypocycloid and Lattice points
Problem 450
Published on Sunday, 15th December 2013, 07:00 am; Solved by 113; Difficulty rating: 100%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 $T(N) = \sum_{R = 3}^N \sum_{r=1}^{\lfloor \frac {R  1} 2 \rfloor} S(R, r)$ be the sum of $S(R, r)$ for R and r positive integers, $R\leq N$ and $2r < R$.
You are given:
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)}
S(3, 1) = (3 + 0) + (1 + 2) + (1 + 0) + (1 + 2) = 10
T(3) = 10; T(10) = 524 ;T(100) = 580442; T(10^{3}) = 583108600.
Find T(10^{6}).