PAMO 2019 Problem 1

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:

• $a_0 = 3$ , $a_1 = 2$ , and $a_2 = 12$ ; and
• $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$ .

Show that $a_n$ is always a strictly positive integer.