KöMaL 2018 A738

(Our original problem in December was wrong. Here we post a new one. You can submit it together with the three problems in January.) Consider the following sequence: $a_1=1$, $a_2=2$, $a_3=3$, and $a_{n+3}=\frac{a_{n+1}^2+a_{n+2}^2-2}{a_n}$ for all integers $n\ge 1$. Prove that every term of the sequence is a positive integer.