Moser's circle problem

This is a somewhat famous problem, known as Moser’s circle problem:

Here’s Wolfram’s formulation:

Determine the number of pieces into which a circle is divided if n points on its circumference are joined by chords with no three internally concurrent.

A less jargony version is:

Draw n points on the circumference of a circle. Connect each pair of points with a straight line segment. What is the maximum number of regions that the circle can be divided into as a function of n?

A few examples:

For n = 2, it has 2 regions,

and for n = 3, it has 4 regions.