Moser's circle problem

Moser's circle problem asks how many regions a circle can be divided into by choosing ${\displaystyle n}$ points along the circumference of the circle and joining each pair of points by a straight line. The greatest possible number of regions with ${\displaystyle n}$ points is given by ${\displaystyle r_{G}={n \choose 4}+{n \choose 2}+1}$, resulting in the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (sequence A000127 in the OEIS). Though the first five terms match the geometric progression ${\displaystyle 2^{n-1}}$, the two sequences differ for ${\displaystyle n\geq 6}$. As Leo Moser noted in 1949, this sequence demonstrates the risk of generalising from only a few observations.