Question 142
Consider sequences $a$ of the form $a = (a_1, a_2, \ldots, a_{20})$ such that each term $a_i$ is either $0$ or $1$. For each such sequence $a$, we can produce a sequence $b = (b_1, b_2, \ldots, b_{20})$, where
$$
b_i = \begin{cases}
a_i + a_{i+1} & i = 1 \\
a_{i-1} + a_i + a_{i+1} & 1 \lt i \lt 20 \\
a_{i-1} + a_i & i = 20
\end{cases}
$$
How many sequences $b$ are there that can be produced by more than one distinct sequence $a$?
$$
b_i = \begin{cases}
a_i + a_{i+1} & i = 1 \\
a_{i-1} + a_i + a_{i+1} & 1 \lt i \lt 20 \\
a_{i-1} + a_i & i = 20
\end{cases}
$$
How many sequences $b$ are there that can be produced by more than one distinct sequence $a$?