Given the group $\mathrm{SL}(2,\mathbb{Q}_p)$ one can describe precisely the number of vertices in the Bruhat-Tits building of distance $k$ from a fixed vertex $v$, where the distance between two vertices $v$ and $v'$ is the minimal number of edges in a path connecting the two. Denoting this quantity by $N_k$, we see in particular that for $\mathrm{SL}(2,\mathbb{Q}_p)$ $$N_k=(q+1)q^{k-1},$$ where $q$ is the size of the residue field.

I am curious if one can come up with an analogous closed formula for $G=\mathrm{SL}(n,\mathbb{Q}_p)$. In the case that $k=1$, this is not difficult and well known by using the lattice interpretation of the building for $\mathrm{SL}$ and counting the number of non-trivial $k$-dimensional subspaces of $\mathbb{F}_p^n$ (one can also use spherical buildings for this and more general groups). For instance, in the case of $n=3$, we have $$N_1=2(q^2+q+1).$$ Continuing the example where $n=3$, to compute $N_2$ is equivalent to finding the number of lattices $\Lambda$ such that if $$\Lambda_0=\mathbb{Z}_p\oplus\mathbb{Z}_p\oplus\mathbb{Z}_p,$$ then $$p^2\Lambda_0\subsetneq \Lambda\subsetneq\Lambda_0,$$ with the additional requirements that $\Lambda$ does not contain $p\Lambda_0$ and that $\Lambda$ is not contained in $p\Lambda_0$ (this ensures that $\Lambda$ is truly distance $2$). However, I have not been able to make the combinatorics of this count work on paper. In general when $n=3$, to compute $N_k$ one would need to count the analogous $\Lambda$s in a chain with $p^2\Lambda_0$ replaced by $p^k\Lambda_0$ subject to the condition that $\Lambda$ does not contain any $p^i\Lambda_0$ and is not contained in any $p^j\Lambda_0$ where $1\le i,j\le k-1$ (to ensure genuine distance $k$).

I am looking to see if a closed formula for $N_k$ is known in the literature or is easy to compute and I am missing the computation. I would be extremely pleased with just the case of $G=\mathrm{SL}(3,\mathbb{Q}_p)$, even if just the case of $n=3$ and $p=2$.