MathNet 2007-part-03 — 185 problems (EN–VI) Question 90

A sequence $\{a_{n}\}_{n \geq 0}$ of real numbers satisfies the recursion $a_{n+1} = a_{n}^{3} - 3 a_{n}^{2} + 3$ for all nonnegative integers $n$. For how many values of $a_{0}$ does $a_{2007} = a_{0}$?