  We study the HNN extension of $\mathbb{Z}^m$ given by the cubing endomorphism
$g\mapsto g^3$, and prove that such groups have rational growth. To do so, we
describe a method of computing the subgroup growth series of the horocyclic
subgroup $\mathbb{Z}^m$ in this family of examples, prove that for all $m$ the
group has rational growth. In the appendix, the subgroup growth series has been
computed for all $m \leq 10$.
