 Directed ladder self-avoiding walk count

On the doubly infinite ladder graph, self-avoiding walks from a fixed graph vertex using only rightward or vertical steps are in bijection with words in $R,V$ containing no consecutive $V$ symbols. Their count is ${\sigma }_n=F_{n+2}$, since ${\sigma }_0=1$, ${\sigma }_1=2$, and ${\sigma }_n={\sigma }_{n-1}+{\sigma }_{n-2}$. Its exponential growth rate is the golden ratio.