BALKAN-SL 2008 N5

Let $(a_n)$ be a sequence with $a_1=0$ and $a_{n+1}=2+a_n$ for odd $n$ and $a_{n+1}=2a_n$ for even $n$ . Prove that for each prime $p>3$ , the number \begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*} for infinitely many values of $n$