2. In the square $R$ (including the boundary) with vertices at $(1997,0)$, $(0,1997)$, $(-1997,0)$, $(0,-1997)$, the number of integer points is ( ) .
(A) 7980011
(B) 7980013
(C) 7980015
(D) 7980017
2. B.

In general, consider the square with vertices at $(N, 0)$, $(0, N)$, $(-N, 0)$, and $(0, -N)$.

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

When $N$ increases to $N+1$, in the first quadrant and on the positive x-axis, the added integer points are $(1, N)$, $(2, N-1)$, $(3, N-2)$, ..., $(N, 1)$, $(N+1,0)$, a total of $N+1$ integer points. By symmetry, a total of $4(N+1)$ integer points are added. Therefore,
$$
a_{N+1}=a_{N}+4(N+1), a_{1}=5 \text{.}
$$

Thus, $a_{N}=a_{1}+4(2+3+\cdots+N)=2 N^{2}+2 N+1$.