For 4 points, there are 8 regions.
For 5 points, there are 16 regions (not 17).
The numbers form the sloan's sequence A027423 (1,2,4,8,16,30,60,96,...)
Eric Weinstein has studied this problem and provided a formula in Mathworld. It seems a little complex for odd values of n. The formula for n even is a simple polynomial.
Note that the formula given in the link below refers to the subdivision of a polygon. For circular division requested, you need to add k, the number of points on the circumference.
Also, the formula starts with n=1 for a triangle, so you need to substitute n=k-2 in the formulas.
For example, for k=4 points, substitute n=k-2=2 in the formula to get 4, but add k to get 8 as your answer for a circular subdivision.