A reccurent sequence

Question

Let $(a_n)$ a sequence such that: $a_1=1$ and $a_2=2$ and $a_3=3$

such that $a_n=\frac{a_{n-1}a_{n-2}+7}{a_{n-3}}$

show that $ a_n \in \mathbb{N} $

I tried to find a particular form of the sequence, but no result

Answer 1

The first 10 terms of $\{a_n\}$ are: $1, 2, 3, 13, 23, 102, 181, 803, 1425, 6322$.

Hint: Show that $a_{n+4}=8a_{n+2}-a_n$.