Question 137
Compute the number of sequences of numbers $a_{1}, a_{2}, \ldots, a_{10}$ such that
$$
\begin{aligned}
& \text{ I. } a_{i}=0 \text{ or } 1 \text{ for all } i \\
& \text{ II. } a_{i} \cdot a_{i+1}=0 \text{ for } i=1,2, \ldots, 9 \\
& \text{ III. } a_{i} \cdot a_{i+2}=0 \text{ for } i=1,2, \ldots, 8
\end{aligned}
$$
$$
\begin{aligned}
& \text{ I. } a_{i}=0 \text{ or } 1 \text{ for all } i \\
& \text{ II. } a_{i} \cdot a_{i+1}=0 \text{ for } i=1,2, \ldots, 9 \\
& \text{ III. } a_{i} \cdot a_{i+2}=0 \text{ for } i=1,2, \ldots, 8
\end{aligned}
$$