HMMT November 2019 T7

Consider sequences $a$ of the form $a=\left(a_{1}, a_{2}, \ldots, a_{20}\right)$ such that each term $a_{i}$ is either 0 or 1 . For each such sequence $a$, we can produce a sequence $b=\left(b_{1}, b_{2}, \ldots, b_{20}\right)$, where

$$b_{i}= \begin{cases}a_{i}+a_{i+1} & i=1 \\ a_{i-1}+a_{i}+a_{i+1} & 1<i<20 \\ a_{i-1}+a_{i} & i=20\end{cases}$$

How many sequences $b$ are there that can be produced by more than one distinct sequence $a$ ?