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Problem 572

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Idempotent matrices

A matrix $M$ is called idempotent if $M^2 = M$.
Let $M$ be a three by three matrix :
$M=\begin{pmatrix}
a & b & c\\
d & e & f\
g &h &i\
\end{pmatrix}$.
Let C($n$) be the number of idempotent three by three matrices $M$ with integer elements such that
$ -n \le a,b,c,d,e,f,g,h,i \le n$.

C(1)=164 and C(2)=848.

Find C(200).

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幂等矩阵

若矩阵$M$满足$M^2 = M$，则称其为幂等矩阵。
令$M$为三阶矩阵：
$M=\begin{pmatrix}
a & b & c\
d & e & f\
g &h &i\
\end{pmatrix}$。
记C($n$)为满足以下条件的三阶幂等矩阵$M$的数目：其元素均为整数且满足
$ -n \le a,b,c,d,e,f,g,h,i \le n$.

已知C(1)=164和C(2)=848。

求C(200)。

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