How many different regions are in a Venn diagram involving n pairwise intersecting sets?

For 2 sets there would be 3 regions. For 3 sets there would be 6 regions if the three intersect at one point, but five if they are lined up in a row. For 4 sets there would be eight regions if they intersect at one point but 7 if they are lined up in a row.

Do I have the definition of pairwise intersecting sets correct?

I'm not sure how to find a formula for this.