Example 4 In a $2 \times n$ grid, color the $2 n$ cells with red and white, each cell being colored with only one color. If it is required that no two adjacent cells are both colored red, how many different coloring methods are there?
Let the number of coloring methods that satisfy the conditions be $a_{n}$.
Let the number of coloring methods where the first column of the $2 \times n$ grid is colored red on top and white on bottom be $b_{n}$.

In $a_{n}$, the first column can be both white or one red and one white, thus, we have $a_{n}=a_{n-1}+2 b_{n}$.

In $b_{n}$, for the second column, it can be both white or white on top and red on bottom, thus, $b_{n}=a_{n-2}+b_{n-1}$.
Therefore, $a_{n}=2 a_{n-1}+a_{n-2}$, and $a_{1}=3, a_{2}=7$.
Hence, $a_{n}=\frac{1}{2}\left[(1+\sqrt{2})^{n+1}+(1-\sqrt{2})^{n+1}\right]$.