Example 7 A password lock's password setting involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex with one of two colors, red or blue, such that for any two adjacent vertices, at least one of the number or color is the same. How many different password settings are there for this type of password lock?
Let the total number of settings that satisfy the conditions be $a_{n}$.
Let the number of settings where the values and colors at points $A_{1}$ and $A_{2}$ are completely the same be $b_{n}$; the number of settings where only one of the value or color at points $A_{1}$ and $A_{2}$ is the same be $c_{n}$; and the number of settings where the values and colors at points $A_{1}$ and $A_{2}$ are completely different be $d_{n}$. Then,
$$
\begin{array}{l}
a_{n}+d_{n}=4 \times 3^{n-1}, \\
a_{n}=b_{n}+c_{n}, b_{n}=a_{n-1} .
\end{array}
$$

Consider the values and colors at points $A_{1}$ and $A_{3}$.
(1) When the values and colors at points $A_{1}$ and $A_{3}$ are completely the same, there are three settings for $A_{2}$, resulting in $3 b_{n-1}$ settings;
(2) When the values and colors at points $A_{1}$ and $A_{3}$ have only one in common, there are two settings for $A_{2}$, resulting in $2 c_{n-1}$ settings;
(3) When the values and colors at points $A_{1}$ and $A_{3}$ are completely different, there are two settings for $A_{2}$, resulting in $2 d_{n-1}$ settings.
Thus, $a_{n}=3 b_{n-1}+2 c_{n-1}+2 d_{n-1}$
$$
=a_{n-2}+8 \times 3^{n-2} \text {. }
$$

It is easy to find that $a_{3}=28, a_{4}=84$.
Therefore, $a_{n}=\left\{\begin{array}{ll}3^{n}+1, & n=2 k+1 \text {; } \\ 3^{n}+3, & n=2 k .\end{array}\right.$