Consider a tiling of the hyperbolic plane defined by a vertex configuration $\Gamma$.
Let in the following be $\Gamma = (3.4)^3$ to have a specific example at hand:
Let the distance spectrum of a "imaginary root" be the sequence of numbers $f(n)$ giving the numbers of vertices on the border of the following sequence $\mathcal{C}_n$ of graphs which are the vertices at some obvious distance $n$ away from the center of the graphs (which is not a vertex), i.e. the vertices on the $n$-th upright square around the center. The sequence starts with $\langle 8, 28, 104,\dots \rangle$.
$\mathcal{C}_1$
$\mathcal{C}_2$
$\mathcal{C}_3$
Compare this spectrum with the distance spectrum of a "real root", i.e. an arbitrary vertex (arbitrary, because the tiling is vertex transitive). The sequence $g(n)$ of numbers of vertices at graph distance $n$ away from a real root starts with $\langle 6, 21, 72 \dots \rangle$.
My question is:
How do these two sequences relate? How can one be derived from the other?