  A hyperplane arrangement in $\mathbb{R}^n$ is a finite collection of affine
hyperplanes. Counting regions of hyperplane arrangements is an active research
direction in enumerative combinatorics. In this paper, we consider the
arrangement $\mathcal{A}_n^{(m)}$ in $\mathbb{R}^n$ given by $\{x_i=0 \mid i
\in [n]\} \cup \{x_i=a^kx_j \mid k \in [-m,m], 1\leq i<j \leq n\}$ for some
fixed $a>1$. It turns out that this family of arrangements is closely related
to the well-studied extended Catalan arrangement of type $A$. We prove that the
number of regions of $\mathcal{A}_n^{(m)}$ is a certain generalization of
Catalan numbers called two parameter Fuss-Catalan numbers. We then exhibit a
bijection between these regions and certain decorated Dyck paths. We also
compute the characteristic polynomial and give a combinatorial interpretation
for its coefficients. Most of our results also generalize to sub-arrangements
of $\mathcal{A}_n^{(m)}$ by relating them to deformations of the braid
arrangement.
