MathNet 2001 — Part 01 Question 116

Let $a_{1}=3$ and define $a_{n+1} = \left(3 a_{n}^{2} + 1\right) / 2 - a_{n}$ for $n \geq 1$. If $n$ is a power of $3$, prove that $a_{n}$ is divisible by $n$.