Eigenvalues · Challenge problem

A sequence satisfies $a_0 = 0$, $a_1 = 1$ and $a_{n+1} = a_n + 2a_{n-1}$ for $n \ge 1$. Writing $\begin{pmatrix}a_{n+1}\\a_n\end{pmatrix} = A\begin{pmatrix}a_n\\a_{n-1}\end{pmatrix}$ and diagonalizing $A$, find $a_{10}$.