 Solution

Consider walks assembled from blocks $E$, $E{N}^k$, and $E{S}^k$ with $k\ge 1$. Their horizontal coordinate increases once in every block, and within each vertical line they move monotonically, so every such walk is self-avoiding. If $c_n$ counts these walks by total length, its ordinary generating function isIts positive dominant singularity is $z=\sqrt{2}-1$, so ${\lim }_n{c}_n^{1/n}=1+\sqrt{2}$. Since $b_n\ge c_n$,