The question is:
How many different undirected graphs are there with n nodes and no parallel edges but may include self-loops?
I've been wracking my brain over this for hours now. Basically, I know that for 1 vertex, there are 2 graphs. For 2 vertices, there are 8 graphs. For 3 vertices, there are 64 graphs. However, I can't connect these numbers without my answer for 1 vertices not working. For example, the equation number of graphs equals $2^{3n-3}$ works for 2 and 3 vertices, but not for 1. Any opinions on the connecting equation?