ILL 1979 Problem 8

The sequence $(a_n)$ of real numbers is defined as follows: $$a_1=1, a_2=2, \text{and} a_n=3a_{n-1}-a_{n-2} , \ \ n \geq 3.$$ Prove that for $n \geq 3$ , $a_n=\left[ \frac{a_{n-1}^2}{a_{n-2}} \right] +1$ , where $[x]$ denotes the integer $p$ such that $p \leq x < p + 1$ .