Question 9
Let $a_n$ be the sequence defined 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
$$
\frac{a_k^j}{a_j^k}
$$
is an integer for all integers $j$ and $k$ with $0 \lt j \lt k$.
$$
a_{n+1} = a_n + 2 \cdot 3^n
$$
for $n \ge 0$.
Determine all rational values of $a_0$ such that
$$
\frac{a_k^j}{a_j^k}
$$
is an integer for all integers $j$ and $k$ with $0 \lt j \lt k$.