Example 7 Divide a circle into $n(n \geqslant 4)$ sectors: $S_{1}, S_{2}, \cdots, S_{n}$. Now, color these $n$ sectors using 5 colors, with the requirement that adjacent sectors have different colors. Try to find the number of coloring schemes.
Let $a_{n}$ be the number of ways to color the sectors $S_{1}, S_{2}, \cdots, S_{n}$. There are two cases: (1) $S_{1}$ and $S_{n-1}$ have the same color; (2) $S_{1}$ and $S_{n-1}$ have different colors. In the first case, $S_{n}$ can be colored with 4 colors, giving a total of $4 a_{n-2}$ coloring schemes; in the second case, $S_{n}$ can only be colored with 3 colors, and the number of coloring schemes for $S_{1}, S_{2}, \cdots, S_{n-1}$ is $a_{n-1}$, thus we have the recursive relation $a_{n}=3 a_{n-1}+4 a_{n-2}$.
And
$$
\begin{array}{c}
a_{2}=5 \times 4=20, a_{3}=5 \times 4 \times 3=60 \\
a_{n}=4^{n}+(-1)^{n} 4 \quad (n=2,3, \cdots)
\end{array}
$$

Solving it, we get
Note that the $a_{n}$ obtained applies to $n=2,3, \cdots$, but does not include the case $a_{1}=5$, because we established the recursive relation based on $n \geqslant 4$.