I want to pair up the integers from 1 to 2n (inclusive) so that the numbers in each pair have a difference of either 1 or n.

For n = 3, I can form the following pairings:

(1 2) (3 4) (5 6)

(1 2) (3 6) (4 5)

(1 4) (2 3) (5 6)

(1 4) (2 5) (3 6)

So there are 4 possible pairings when n = 3.

In general (for any n) how many possible pairings are there?