Literally, the only way to have n pairs is to remove 1 from 2n+1.
However, I suspect that Ady really wants us to have (2n + 1 - x)/2 pairs.
In this case, x can be any odd number from 1 to 2n-3. The x numbers can be removed from the low end of the sequence, or from the high end, or from the exact middle.
If x = 2n-1, then there is only one remaining pair, and arguably the puzzle condition is not met.
If x = 2n+1, then there are no remaining pairs, and arguably the puzzle condition is not met.
An earlier solution suggested n + 1, but this fails if n is odd.