Example 4 On the line segment $A_{1} A_{n}(n \geqslant 2)$, there are $n$ points $A_{1}, A_{2}, \cdots, A_{n}$. Now, some of these points (or the empty set) are painted red, so that the colored points are not adjacent. Find the number of ways to color.
Let the number of coloring methods that satisfy the conditions be $a_{n}$.
When $n=2$, there are (red, none), (none, red), (none, none), a total of 3 cases, $a_{2}=3$;

When $n=3$, there are (red, none, none), (none, red, none), (none, none, red), (red, none, red), (none, none, none), a total of 5 cases, $a_{3}=5$.
When $n \geqslant 4$, consider point $A_{n}$.
If $A_{n}$ is uncolored, then the coloring methods of the first $n-1$ points are also the coloring methods of $n$ points, there are $a_{n-1}$ ways;

If $A_{n}$ is red, then $A_{n-1}$ must be uncolored. Thus, the coloring methods of the first $n-2$ points are also the coloring methods of $n$ points, there are $a_{n-2}$ ways. Hence, we get