MathNet 2011 Question 17

$a_0=1$, $a_1=3$ and $a_{n+2}=1+\left\lfloor a_{n+1}^2/a_n\right\rfloor$ for all $n\ge0$. Show that $a_{n+2}a_n-a_{n+1}^2=2^n$ for all integers $n$.