Directed ladder self-avoiding walk count
= Directed ladder self-avoiding walk count
{title2=$\sigma_n=F_{n+2}$}
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>.