5. $(44,19)$ Since the number of lattice points in the region $0 \leqslant x \leqslant k, 0 \leqslant y \leqslant k$ is $(k+1)^{2}$, and $44^{2}=19362006$, the x-coordinate or y-coordinate of the point with number 2006 is 44. Since 44 is even, the point should be counted from $(0,44)$ to the right. Since $2006-44^{2}=70>44$, the x-coordinate of the point with number 2006 is 44, and the y-coordinate is $44-(70-45)=19$. Therefore, the coordinates of the point with number 2006 are $(44,19)$.
5. In the Cartesian coordinate system, grid points are numbered as follows: $(0,0)$ is No. 1, $(1,0)$ is No. 2, $(1,1)$ is No. 3, $(0,1)$ is No. 4, $(0,2)$ is No. 5, $(1,2)$ is No. 6, $(2,2)$ is No. 7, $(2, 1)$ is No. 8, $(2,0)$ is No. 9, (as shown in Figure 7-1). Following the order of the arrows in the figure, the coordinates of the 2006th point are . $\qquad$