Max/Min Sums From Permutations

[Leroy Quet]

Say we have a permutation of <1,2,3,...,n>,
<p(1),p(2),p(3),...,p(n)>.

Next, take the sum S of products formed from the permutation,
S = p(1)*p(2) + p(2)*p(3) + p(3)*p(4) +... + p(n-1)*p(n).

(So, every term besides p(1) and p(n) appear in 2 products, p(1) and p(n) 
appear once each.)

For a given n, what is the maximum possible and minimum possible S?


I get by hand, and not necessarily correctly, that the sequence of 
maximums begins
1(or 0), 2, 9, 23, 50,...

I get the minimums (again, most likely incorrectly) as
1(or 0), 2, 5, 12, 22,...

Could someone compute the real sequences and submit them to the OEIS?

thanks,
Leroy Quet