Problem 596
Number of lattice points in a hyperball
Let T($r$) be the number of integer quadruplets $x$, $y$, $z$, $t$ such that $x^2+y^2+z^2+t^2 \le r^2$. In other words, T($r$) is the number of lattice points in the four-dimensional hyperball of radius $r$.
You are given that T(2) = 89, T(5) = 3121, T(100) = 493490641 and T(104) = 49348022079085897.
Find T(108) mod 1000000007.
超球中的格点数目
记T($r$)为满足$x^2+y^2+z^2+t^2 \le r^2$的整数四元组$x$，$y$，$z$，$t$的数目。换言之，T($r$)是半径为$r$的四维超球中格点的数目。
已知T(2) = 89，T(5) = 3121，T(100) = 493490641以及T(104) = 49348022079085897。
求T(108) mod 1000000007。