HMMT 2010 U33

Let $a_{1}=3$, and for $n>1$, let $a_{n}$ be the largest real number such that

$$4\left(a_{n-1}^{2}+a_{n}^{2}\right)=10 a_{n-1} a_{n}-9$$

What is the largest positive integer less than $a_{8}$ ?