Problem

Source: India Postal Coaching 2015

Tags: Sequence, induction, Recurrence, number theory



The sequence $<a_n>$ is defined as follows, $a_1=a_2=1$, $a_3=2$, $$a_{n+3}=\frac{a_{n+2}a_{n+1}+n!}{a_n},$$$n \ge 1$. Prove that all the terms in the sequence are integers.