JBMO-SL 2020 A2

Consider the sequence $a_{1}, a_{2}, a_{3}, \ldots$ defined by $a_{1}=9$ and

$$a_{n+1}=\frac{(n+5) a_{n}+22}{n+3}$$

for $n \geqslant 1$.

Find all natural numbers $n$ for which $a_{n}$ is a perfect square of an integer.