Suppose we have an undirected graph with integer valued nodes where $0<|i-j|\le 2$ implies nodes $i$ and $j$ are connected. Let $c_n$ be the number of self-avoiding walks on this graph of length $n$ starting at origin. Define the connective constant as

$$\mu = \lim_{n\to \infty} c_n^{\frac{1}{n}}$$

What is known about $\mu$? This quantity seems to be related to the transition temperature of an Ising model on such graph, has such model been studied?