We study rational circular billiards. By viewing the trajectory formed after each reflection point to another inside the circle as the number of circle divisions into regions we derive a general formula for the number of division regions after each reflection... This will give rise to an integer division sequence. Restricting to the special cases $\vartheta =\frac{q}{2q+1}\cdot 2\pi$ we show that the number of regions after each reflection $n$ is beautifully related to Gauss 's arithmetic series. (read more)