Since there are only $N^3$ possibilities for $(a_{k-1},a_k,a_{k+1})\pmod{N}$, it follows by PHP that $a_n$ is eventually periodic. In fact, since the recursion is invertible, it is totally periodic. Extending the definition to all integers, we see that $a_{-1}=10-2^3-1^2=1$ and $a_{-2}=2-1^3-1^2=0$. Therefore, if $T$ is the period of $a_n\pmod N$, $N|a_{T-2}$.