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