HMMT 2007 F8

Compute the number of sequences of numbers $a_{1}, a_{2}, \ldots, a_{10}$ such that

$$\begin{aligned} & \text { I. } a_{i}=0 \text { or } 1 \text { for all } i \\ & \text { II. } a_{i} \cdot a_{i+1}=0 \text { for } i=1,2, \ldots, 9 \\ & \text { III. } a_{i} \cdot a_{i+2}=0 \text { for } i=1,2, \ldots, 8 \end{aligned}$$