Infinite Sequences and Series · Challenge problem

A sequence is defined by $a_0=0$, $a_1=1$ and $a_{n+2}=a_{n+1}+2a_n$ for $n\ge0$, so it begins $0,1,1,3,5,11,21,\dots$. Find $\displaystyle\sum_{n=0}^{\infty}\frac{a_n}{4^n}$.