Consider $n$ points $P_{1}, P_{2}, \cdots, P_{n}$ arranged sequentially on a straight line, each point is colored with one of white, red, green, blue, or purple. If for any two adjacent points $P_{i}, P_{i+1}(i=1,2, \cdots, n-1)$, either these two points are the same color, or at least one of them is white, then it is called a good coloring. Find the total number of good colorings.
Let $a_{n}$ denote the number of good colorings of $P_{n}$ where $P_{n}$ is colored white, and $b_{n}$ denote the number of good colorings of $P_{n}$ where $P_{n}$ is not colored white. Then, $c_{n}=a_{n}+b_{n}$ represents the total number of good colorings. From the definitions of $a_{n}$, $b_{n}$, and $c_{n}$, we have: $a_{n}=c_{n-1}$, $b_{n}=4 a_{n-1}+b_{n-1}$, so $c_{n}=a_{n}+b_{n}=c_{n-1}+b_{n-1}+4 a_{n-1}=c_{n-1}+b_{n-1}+4 c_{n-2}$. Noting that $b_{n}=c_{n}-a_{n}=c_{n}-c_{n-1}$, substituting this into the above equation gives $c_{n}=2 c_{n-1}+3 c_{n-2}$.
It is easy to see that: $c_{1}=5, c_{2}=13$, and using the characteristic root method, we get $c_{n}=\frac{1}{2}\left(3^{n+1}+(-1)^{n+1}\right)$.