32. Let $a_{i}=0,1(i=1,2, \cdots, n)$, find the number of sequences $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ that satisfy $a_{1} \leqslant a_{2}, a_{2} \geqslant a_{3}, a_{3} \leqslant a_{4}$, $a_{4} \geqslant a_{5}, \cdots$.
32. Let $f_{n}$ be the number of sequences sought, then $f_{1}=2, f_{2}=3, f_{3}=5$.

When $n$ is odd, divide the sequences into two categories: (1) $a_{n}=1$, then $a_{n-1}=1$, so there are $f_{n-2}$ sequences that meet the requirement; (2) $a_{n}=0, a_{n-1}$ can be 1 or 0; when $a_{n-1}=1$, there are $f_{n-2}$ sequences; when $a_{n-1}=0$, $a_{n-2}$ must be 0, there are $f_{n-3}$ sequences, thus $f_{n}=2 f_{n-2}+f_{n-3}(n \geqslant 4)$.
When $n$ is even, the same recurrence relation can also be derived.
From the derived recurrence relation and induction, it can be proven that $f_{n}=f_{n-1}+f_{n-2}$, so $f_{n}=F_{n+2}$, where $F_{n}$ is the Fibonacci sequence.