Extending the Olympic rings puzzle

Below is a puzzle. The objective is to place 7 dots such that each circle contains the same number of dots.

The solution is not trivial, but can be found quickly, as seen here. But wherever we see numbers or structure, we want to stretch and bend, in pursuit of other interesting puzzles. In this case, we can ask "what about placing n dots?" for various n. We've already found a solution for n = 7. What about other values of n? Certainly n = 0 works, and n = 1 does not. Is there a closed formula for which n works? Is there a recursive formula which is perhaps more intuitive? In our puzzle we have 5 rings, but what if we had some other number? What if the rings were arranged in another way, such as a circle? What if we not only allowed circles to intersect, but also allowed smaller circles to lie inside larger ones? Generalizing a bit further, what about an arbitrary Venn diagram? And what makes such a puzzle interesting? Is it the number of solutions for a particular number of rings and dots? Does it have something to do with solutions being asymmetric?