shoe lacings

[N. J. A. Sloane]

Today's New York Times (p. F3) has a report about
Burkard Polster's enumeration of the ways to
lace a shoe with 2 pairs of eyelets that is apparently
in the current issue of Nature. For 6 pairs of
eyelets there are "over 43200" ways.

Naturally I'm interested in the sequence that gives
the number of ways if there are n pairs of eyelets.
This begins 1, 2, but the next term is unclear
because I don't know what the full set of rules is.
In fact there will probably be several versions
of the sequence.  Can anyone help?

 From the NYT article it seems that the lace must 
pass through each eyelet exactly one, must begin and end
at the extreme pair of eyelets, and cannot pass though three
adjacent eyelets that are in a line.  But perhaps there are other
less obvious restrictions.

If there are no hidden restrictions i get 23 for the third term.

NJAS