TUYMAADA 1999 Problem 7

A sequence of integers $a_0,\ a_1,\dots a_n \dots$ is defined by the following rules: $a_0=0,\ a_1=1,\ a_{n+1} > a_n$ for each $n\in \mathbb{N}$ , and $a_{n+1}$ is the minimum number such that no three numbers among $a_0,\ a_1,\dots a_{n+1}$ form an arithmetical progression. Prove that $a_{2^n}=3^n$ for each $n \in \mathbb{N}.$