What other sequences solve this equation?

[Edwin Clark]

On Wed, 10 May 2006, Kimberling, Clark wrote:

> In this query, a(n) and b(n) denote increasing complementary sequences,
> such as (1,3,5,7,...) and (2,4,6,8...), as well as pairs of Beatty
> sequences.
>  
> The question is, how can we account for all solutions of the equation 
>  
> a(b(n)) - b(a(n)) = 1 ?
>  

A finite version of the question:

I decided to try to find permutations f and g of {0,1,...,n-1} so
that g(f(i)) = f(g(i)) + 1 (mod n) for i from 0 to n-1.
I found for n = 1,...,7, the number a(n) of such pair is:

    1, 0, 9, 0, 200, 0, 8820

which if we leave out the 0s agrees up to the last term with
the sequence http://www.research.att.com/~njas/sequences/A052145
which is given by the formula 2*n*n!/(n+1) for n odd. But since
I don't have the reference at hand, I'm not sure it's from the
same problem I just stated--but with such strange numbers 
how could it not be? Also my method is too crude to calculate
more terms easily.