MathNet 2013 Question 90

Given an integer $n \ge 2$, let $a_n$, $b_n$, $c_n$ be integer numbers such that $(\sqrt[3]{2}-1)^n = a_n + b_n\sqrt[3]{2} + c_n\sqrt[3]{4}$. Show that $c_n \equiv 1 \pmod{3}$ if and only if $n \equiv 2 \pmod{3}$.