Example 8 A point whose both coordinates are integers is called an integer point. If $n$ is a non-negative integer, then, in the region
$$
\{(x, y) \mid | x|+| y | \leqslant n\}
$$

how many integer points are there?
Explanation: As shown in Figure 2, it is evident that the region is a square.

Let there be $a_{\mathrm{n}}$ integer points in it, and obviously $a_{0}=1$.

When $a_{n}$ increases to $a_{n+1}$, the points in the first quadrant and on the positive half-axis of the $x$-axis that will be added are those with the sum of their coordinates equal to $n+1$: $(1, n),(2, n-1), \cdots,(n, 1),(n+1,0)$. By symmetry, we have
$$
\begin{array}{l}
a_{n+1}-a_{n}=4(n+1) . \\
\text { Therefore, } a_{n}=\left(a_{n}-a_{n-1}\right)+\left(a_{n-1}-a_{n-2}\right)+ \\
\cdots+\left(a_{1}-a_{0}\right)+a_{0} \\
=4 \sum_{i=1}^{n} i+1=2 n^{2}+2 n+1 .
\end{array}
$$