IMO 1988 - Shortlisted problems Question 1

(BUL 1) An integer sequence is defined by $$a_n=2a_{n-1}+a_{n-2}\quad(n\gt 1),\qquad a_0=0,\quad a_1=1.$$ Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.