Partially directed self-avoiding walk

On the square lattice, a self-avoiding walk using north, east and west steps but no south steps. At each horizontal height, self-avoidance forces one monotone horizontal run. Conversely, such runs separated by north steps always give self-avoiding walks, because every new run is at a previously unvisited height.

A horizontal run has generating function $R(z)=(1+z)/(1-z)$. Decomposing each walk into $H(NH{)}^k$ gives $A(z)=R(z)/(1-zR(z))=(1+z)/(1-2z-z^2)$. Thus $a_0=1,a_1=3$ and $a_n=2a_{n-1}+a_{n-2}$. The exponential growth rate is $1+\sqrt{2}$, furnishing a strict lower bound greater than two for the connective constant of all square-lattice self-avoiding walks.