OME 2012 Problem 5

A sequence $(a_n)_{n\ge 1}$ of integers is defined by the recurrence $$a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2.$$ Prove that all terms of the sequence are integers and find an explicit formula for $a_n$ .