MathNet 2007-part-02 — 187 problems (EN–VI) Question 17

The sequence $\{a_n\}$ is defined by $a_1 = 0$ and $(n+1)^3 a_{n+1} = 2n^2(2n+1)a_n + 2(3n+1)$ for all integers $n \ge 1$. Show that infinitely many members of the sequence are positive integers.