Each number covers at most $4$ others, $5$ total including itself. There are $16$ sequences so we need to choose at least $4$. Checking to see if this works, we find that choosing the sequences $(0,0,0,0), (1,0,0,0), (0,1,1,1), (1,1,1,1)$ works, so $\boxed{4}$ is the answer.