Count number of binary strings that satisfy some conditions?

Question

Let b be binary string of length r < n, b doesnt contain any adjacent 0s or 1s. Count number of binary string of length n that if we repeatedly apply these operations 1) 11 =1, 2) 00=0. b doesnt contain adjacent 0s or 1s. For example r=2, n=3, b=01

The strings that result b after those operations are 001, and 011 so the answer is 2

Answer 1

Since this is an old question which is a bit ambiguous, I will reformulate it and post it again. I will supply an answer as well.