As to the convergenge to zero: note that the convergents of the [continuous fraction for $\pi$][1] 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][2].

[1]:http://oeis.org/A001203
[2]:http://oeis.org/A002485