Into the 4th dimension

Consider a sphere whose square of its radius is $n$ and centered at $(0,0,0,0)$ in the $4^{th}$ dimension space. For a given $n$, the sphere touches a certain number of integer lattice points. For instance, if $n=7$, it touches the point (1,2,1,1) Let $t(n)$ the number of integer lattice points touched by the sphere with square of radius $n$
Find $\sum _{k=0}^{100}t({10}^{17}+k)$

You are given $\sum _{k=0}^{100}t(1000+k)=1042664$

[My timing: 9 sec]