In mathematics, a Self-Avoiding Walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. (see Wikipedia )

The question of how many possible n-step SAWs are available on any given lattice is an open problem. It's possible to calculate the upper and lower bounds for the number of walks :

[…] but still the only known way to get an exact tally is to actually trace out all the n-step walks and count them.

Source : American Scientist , Volume 86, Number 4, pages 314–319

For a recent study into a possible mathematical model, see : Theophilus Agama (2021) On a function modeling an l-step self avoiding walk, AKCE International Journal of Graphs and Combinatorics.