integrality of terms of a sequence

Question

The sequence $(a_n)_{n \ge 1}$ is defined by $a_i=i$ for $i=1,2,3$ and satisfies $$a_{i}=\dfrac{a_{i-1}a_{i-2}+7}{a_{i-3}}, i \ge 4.$$ The question is to prove that all terms of the sequence are integers.

Thanks in advance for any hints.

Answer 1

As mentioned in the comments by @Ronald Blaak your sequence can actually be simplified into a simpler sequence $a_i=8a_{i-2}-a_{i-4}$. To prove this you can use induction: the base case is trivial, suppose now that you know it is true for all $j\le{}i$ then multiplying both sides by $a_{i-1}$ and dividing with $a_{i-2}$ we get $a_{i+1}=\frac{7+a_ia_{i-1}}{a_{i-2}}=\frac{8a_{i-2}a_{i-1}}{a_{i-2}}-\frac{a_{i-4}a_{i-1}+7}{a_{i-2}}$, but $a_{i-4}a_{i-1}=a_{i-2}a_{i-3}+7$ and thus $a_{i+1}=\frac{a_{i-4}a_{i-1}+7}{a_{i-2}}=8a_{i-1}-a_{i-3}$ which completes the proof.