Counting all unfolded self-avoiding walks on a finite lattice strip of width three

Abstract

By extending the method of the author’s earlier paper, a purely recursive approach is devised for counting the total number of unfolded self-avoiding walks terminating along the line $x=n$, within the finite lattice strip $\left\{0,1,\dots ,n\right\}\times \left\{0,1,2,3\right\}$ of width three. This approach yields a generating function for the sequence of numbers just described, without the need for any exact enumeration, usually entailed with an application of the transfer matrix algorithm. As a result of determining this generating function, the total number of unfolded walks within the lattice strip can then be easily computed.