Arranging 2N numbers

[N. J. A. Sloane]

A correspondent asks:

Arrange the numbers 1,1,2,2,3,3,4,4.....15,15,16,16 in such a way that
there is one number between the ones, two numbers between the twos and
so on up to 16 so that there are 16 numbers between the 16s.

Here is a solution found by a high-school student:
16, 14, 12, 10, 13, 5, 6, 4, 15, 11, 9, 5, 4, 6, 10, 12, 14, 16, 13, 8, 9, 11,
7, 1, 15, 1, 2, 3, 8, 2, 7, 3

So I ask:
Suppose we take the numbers 1,1,2,2,3,3,...,N,N and ask the same question.

For what values of N is it possible?
(N=1, 2, no!  N=3, 4, 16, yes!)
Is there a recursive construction?
This must be an old problem.  
What if they are arranged around a circle? (And "between" means along the shorter path)

Neil Sloane