VN-TST 2011 Problem 4

Let $\langle a_n\rangle_{n\ge 0}$ be a sequence of integers satisfying $a_0=1, a_1=3$ and $a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.$ Prove that $a_n\cdot a_{n+2}-a_{n+1}^2=2^n$ for every natural number $n.$