1. Use $A$, $B$, and $C$ to form a permutation of length $n$ (with $n$ letters), but it is not required that all three letters appear in the permutation. If no two $A$s are adjacent, how many such permutations are there?
Let the number of permutations of length $n$ that satisfy the condition be $a_{n}$. Then,
$$
a_{n}=2 a_{n-2}+2 a_{n-1}(n \geqslant 3) .
$$

It is easy to see that $a_{1}=3, a_{2}=8$.
Solving this, we get
$$
\begin{aligned}
a_{n}= & \frac{(2+\sqrt{3})(1+\sqrt{3})^{n}}{2 \sqrt{3}}+ \\
& \frac{(-2+\sqrt{3})(1-\sqrt{3})^{n}}{2 \sqrt{3}} .
\end{aligned}
$$