  A growing self-avoiding walk (GSAW) is a walk on a graph that is directed,
does not visit the same vertex twice, and has a trapped endpoint. We show that
the generating function enumerating GSAWs on a half-infinite strip of finite
height is rational, and we give a procedure to construct a combinatorial finite
state machine that allows one to compute this generating function. We then
modify this procedure to compute generating functions for GSAWs under two
probabilistic models. We perform Monte Carlo simulations to estimate the
expected length and displacement for GSAWs on the quarter plane, half plane,
full plane, and half-infinite strips of bounded height for which we cannot
compute the generating function. Finally, we prove that the generating
functions for Greek key tours (GSAWs on a finite grid that visit every vertex)
on a half-infinite strip of fixed height are also rational, allowing us to
resolve several conjectures.
