Writing $a_n=\frac{p_n}{q_n}$ in reduced form, we get $a_{n+1}=\frac{q_n}{q_n \cdot \left(2\left\lfloor \frac{p_n}{q_n}\right\rfloor+1\right)-p_n}$ which is still in reduced form, hence $p_{n+1}=q_n$ and so $a_n=\frac{p_n}{p_{n+1}}$ where $p_1=p_2=1$ and $p_{n+1}=p_n \cdot \left(2\left\lfloor \frac{p_{n-1}}{p_n}\right\rfloor+1\right)-p_{n-1}$. But this is the famous Stern-Brocot sequence. In particular, amazingly, every positive rational number occurs exactly once in the sequence $a_i$.