Spoiler for Long shot

(2^{n}-1) + ((n^{2}-n)/2) + ((3^{n+1} - 2^{n+3} + 6 + (-1)^{n+1})/12)

Validated out to n=5 so far...I'm pretty sure on the first two parts of that formula...it's the last/third that I'm a little questionable on...and if that's correct, here are the values up to 9:

n=1: 1

n=2: 4

n=3: 12

n=4: 31

n=5: 81

n=6: 218

n=7: 610

n=8: 1753

n=9: 5127