As to the convergenge to zero: note that the convergents of the continuous fraction for $\pi$ provide a rational approximation $|\pi - p_n/q_n| < 1/q_nq_{n+1}$ so that $\sin p_n\to 0$.
The sequence of numerators $0, 1, 3, 22, 333, 355,\dots$ is OEIS' A002485.
