MathNet 2010 — Part 05 Question 152

Consider the sequence $a_{1} = 1$ and $a_{n+1} = \frac{3 a_{n}^{2} + 1}{2} - a_{n}$ for $n = 1, 2, \ldots$. Prove that if $n$ is a power of $3$ then $n$ divides $a_{n}$.