AT-OEMO-FED1 2014 Problem 3

Let $a_n$ be a sequence defined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$ . Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$ .