8. $[\mathbf{5}]$ Compute the number of sequences of numbers $a_{1}, a_{2}, \ldots, a_{10}$ such that
I. $a_{i}=0$ or 1 for all $i$
II. $a_{i} \cdot a_{i+1}=0$ for $i=1,2, \ldots, 9$
III. $a_{i} \cdot a_{i+2}=0$ for $i=1,2, \ldots, 8$.
Answer: 60 . Call such a sequence "good," and let $A_{n}$ be the number of good sequences of length $n$. Let $a_{1}, a_{2}, \ldots, a_{n}$ be a good sequence. If $a_{1}=0$, then $a_{1}, a_{2}, \ldots, a_{n}$ is a good sequence if and only if $a_{2}, \ldots, a_{n}$ is a good sequence, so there are $A_{n-1}$ of them. If $a_{1}=1$, then we must have $a_{2}=a_{3}=0$, and in this case, $a_{1}, a_{2}, \ldots, a_{n}$ is a good sequence if and only if $a_{4}, a_{5}, \ldots, a_{n}$ is a good sequence, so there are $A_{n-3}$ of them. We thus obtain the recursive relation $A_{n}=A_{n-1}+A_{n-3}$. Note that $A_{1}=2, A_{2}=3, A_{3}=4$. Plugging these into the recursion eventually yields $A_{10}=60$.