## When 361..1*3611...1 is square number ?

Let $a_{n}\subset\mathbb{N}$ define as

$a_{1}= 36\\ a_{2}= 361\\ a_{3}= 3611\\ a_{4}= 36111\\ a_{5}= 361111\\ \cdots\\ a_{n}= 36111\cdots 11\\ \cdots$

When $a_{m}* a_{n}\; (m < n)$ is square number ?