Counting admissible MacMahon parameters for 3x3 squares

The number of admissible MacMahon parameter pairs for line sum $3e$ is

Proof. Substituting $p = a - e$ and $q = c - e$, the four inequalities read $|p+q| \le e$ and $|p-q| \le e$, and since $\max(|p+q|,|p-q|) = |p| + |q|$ this is exactly $|p| + |q| \le e$: the $\ell_{1}$ ball of radius $e$ in $\mathbb{Z}^{2}$. On the sphere $|p| + |q| = k$ there are $4k$ lattice points for $k \ge 1$ and one for $k = 0$, so the ball has

points.

Equivalently one may sum over $a$: for fixed $a \in [0,2e]$ the admissible $c$ form the interval $[\,|a-e|,\ \min(a+e,\,3e-a)\,]$, which has $2a+1$ elements when $a \le e$ and $4e-2a+1$ elements when $a \ge e$; summing gives $(e+1)^{2} + e^{2} = 2e^{2} + 2e + 1$.

Formalization Note paramCount e is the cardinality of the finset paramSet e, defined by filtering the box $[0,2e] \times [0,2e]$ — the bounds $a, c \le 2e$ are implied by admissibility, so this is lossless.