I'm interested in estimating the number of binary strings $(a_1, a_2, \dots, a_n)$ that avoid a specific type of structure. A specific example is the following: for any $k$, $(a_{k-3}, a_{k-2}, a_k, a_{k+2})\ne (0,0,1,0)$. One can visualize these objects as "combs", and when shifting a comb along the sequence we are not allowed to hit $(0,0,1,0)$.

If one were to remove the $a_{k-3}$ the problem has a simple solution with recursion, and asymptotics are not hard to obtain. I suspect that this can be bounded by probabilistic arguments. My guess is the answer should be $\Theta(c^n)$ for some constant $c$, and for the specific comb mentioned above I suspect $c\approx 15/8$. I'm more interested in a lower bound than an upper bound, for what it's worth.

I would appreciate if anyone could direct me to a solution to this and/or any literature on this type of problem.