Problem: An $n$-term sequence $(x_1, x_2, \dots, x_n)$
in which each term is either $0$ or $1$ is called a binary sequence of length $n$.
Let $a_n$ be the number of binary sequences of length $n$ containing
no three consecutive terms equal to $0$, $1$, $0$ in that order.
Let $b_n$ be the number of binary sequences of length $n$ that
contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order.
Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.

Solution: Consider the map from
sequences of the latter form to sequences of the first form by
\[ (y_1, \dots, y_{n+1}) \mapsto (y_1 + y_2, y_2 + y_3, \dots, y_n + y_{n+1}). \]
It is $2$-to-$1$. The end.