Example 6. $N$ is a positive integer, in the square $R$ with vertices at $(N, 0)$, $(0, N)$, $(-N, 0)$, $(0, -N)$ (including the boundary), how many integer points are there?
Let the number of integer points in the square $R$ (including the boundary) be $a \times$.

When $N=1$, the integer points are (1,0), (0,1), (-1,0), (0,-1), and (0,0), totaling 5 points, so $a_{1}=5$.

When $N$ increases to $N+1$, in the first quadrant and the positive half-axis of the $x$-axis, the added integer points are $(1, N), (2, N-1), (3, N-2), \cdots, (N, 1), (N+1,0)$, a total of $N+1$ integer points. By symmetry, this adds $4(N+1)$ integer points.

Thus, $a_{N+1}=a_{N}+4(N+1)$.
From $\left\{\begin{array}{l}a_{1}=5, \\ a_{N+1}=a_{N}+4(N+1)\end{array}\right.$

we can derive
$$
a_{N}=a_{1}+4(2+\cdots+N)=2 N^{2}+2 N+1,
$$

so the square $R$ contains $2 N^{2}+2 N+1$ integer points.