Let a ($D_4$-)symmetric configuration-$\Gamma$ complex be a planar graph with the symmetry group of the square, i.e. the octic group $D_4$, in which the vertices with maximal degree have the same vertex configuration $\Gamma$. For the sake of simplicity, assume that $\Gamma$ has the form $(3.n)^m$ with $n>3$ and $m>2$.
Example: $(3.4)^3$
These are the most simple examples $\mathcal{C}_2$, $\mathcal{C}_3$ which are symmetric $(3.4)^3$ complexes:
$\mathcal{C}_2$
$\mathcal{C}_3$
[Side question: For which vertex configurations $\Gamma$ do $D_4$-symmetric $\Gamma$ complexes exist?]
Let $|\mathcal{C}_n|$ be the number of vertices of $\mathcal{C}_n$. We count $|\mathcal{C}_2| = 36$ and $|\mathcal{C}_3| = |\mathcal{C}_2| + 4 \cdot 26 = 140$ .
In general we have $|\mathcal{C}_{n+1}| = |\mathcal{C}_n| + 4 \cdot f(n)$ with $f(2) = 7$ and $f(3) = 26$ for $\Gamma = (3.4)^3$, where $\mathcal{C}_1$ is this graph (not a symmetric configuration complex):
What I am looking for is a generating function for either $|\mathcal{C}_n|$ or $f(n)$, either for the specific vertex configuration $(3.4)^3$, or general $(3.n)^m$, or even for arbitrary vertex configurations.