MathNet 2024-part-01 — 168 problems (EN–VI) Question 97

Given is a sequence $a_1, a_2, \dots$, such that $a_1 = 1$ and $a_{n+1} = \frac{9a_n+4}{a_n+6}$ for any $n \in \mathbb{N}$. Which terms of this sequence are positive integers?