Consider walks assembled from blocks $E$, $EN^k$, and $ES^k$ with $k\geq1$. 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> is
$$
\sum_{n\geq0}c_nz^n
=\frac1{1-\left(z+2\sum_{k\geq1}z^{k+1}\right)}
=\frac{1-z}{1-2z-z^2}.
$$
Its positive dominant singularity is $z=\sqrt2-1$, so $\lim_nc_n^{1/n}=1+\sqrt2$. Since $b_n\geq c_n$,
$$
\boxed{\kappa\geq1+\sqrt2>2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 204 / 1 / c / Solution

{title2=$A(z)=\sum_{n\geq0}a_nz^n$}
{wiki=Generating_function}

The ordinary generating function of a sequence $(a_n)_{n\geq0}$ is the <formal power series> $A(z)=\sum_{n\geq0}a_nz^n$. Algebraic identities for $A$ encode recurrences and counting constructions for the coefficients.

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