Wow, that's good reasoning!

I though it would be more difficult to correct those problems.
This is an interesting variation.

So you are right, we do get
(n^4 - 2n^3 + 3n^2 - 2n + 8)/8 - (n^3 - 5n^2 + 6n)/2
= (n^4 -6n^3 + 23n^2 -26n +8)/8

For n=2 to 14, this is

2, 7, 18, 41, 85, 162, 287, 478, 756, 1145, 1672, 2367, 3263, ..

I checked on a drawing for n=5 and there are 41 regions.

Good work.

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PS: After all this, (that is after I got your answer),
I searched "7,18,41,85" on Google
and found it in the Encyclopedia of Integer Sequences
(It is not in the printed version I have)

http://www.research.att.com/~njas/sequences/A055503
where the following reference is given:
L. Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, p. 72; and Problem 8, p. 74.