Closed-form formula for the number of exterior intersection points when n is even

Determine an explicit closed-form formula for N_{e,n}, the number of intersection points lying outside the unit circle generated by the lines through all vertex pairs of a regular n-gon, for even integers n. For odd n, this quantity is known to satisfy N_{e,n} = n(2n^3 − 15n^2 + 34n − 21)/24, but an analogous formula for even n is currently unknown due to the lack of a concurrency-based argument outside the unit circle that parallels the Poonen–Rubinstein approach for interior points.

Background

The paper studies the distribution and enumeration of intersection points formed by the lines determined by all pairs of vertices of a regular n-gon (the clique-arrangement). Intersection points are partitioned into interior points (inside the unit circle) and exterior points (outside the unit circle).

Poonen and Rubinstein previously derived formulas for the number of interior intersection points N_{i,n} and for regions, leading to exact counts for many combinatorial features inside the polygon. By combining their results with the total number of points N_n and subtracting the n vertices, one obtains the exterior count N_{e,n} in specific cases.

The authors note that a closed-form expression for N_{e,n} is known when n is odd: N_{e,n} = n(2n^3 − 15n^2 + 34n − 21)/24. However, for even n, the analogous formula is not known because the concurrency and similar-triangle arguments that work inside the unit circle do not extend to the exterior configuration.