Integer Sequences from Circle Divisions in Rational Billiards

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.