Question 4
A binary sequence of length $n$ is a sequence $(x_1,\ldots,x_n)$ whose terms are $0$ or $1$. Let $a_n$ count such sequences with no consecutive pattern $010$. Let $b_n$ count such sequences with neither consecutive pattern $0011$ nor consecutive pattern $1100$. Prove that $b_{n+1}=2a_n$ for every positive integer $n$.