Discrete Math

Find a recurrence and appropriate initial conditions for the number of binary strings of length n which every 0 is immediately preceded or followed by another 0, but not both.

For example, these are OK: 00 100 111 (we dont promise that 0 appears) 1001 001100

These are not OK: 0 (not preceded or followed by 0) 000 (both preceded and followed by 0) 001101 (last 0 not preceded or followed by 0)

You must justify your answer. (In particular, say where every term in your recurrence came from, and explain why no additional terms are needed.)