Explicit Apéry set formula for Γ_6

Let $\Gamma_6$ be generated by $s_n=\frac{n}{2}[2a+(n-1)d]$ for $1\leq n\leq 6$, where $a,d>0$ and $\gcd(a,d)=1$. For $n\in\mathbb{N}$, write

$$n=15s_{4n}+t_{4n},\qquad 0\leq t_{4n}\leq 14,$$ $$t_{4n}=10s_{3n}+t_{3n},\qquad 0\leq t_{3n}\leq 9,$$ $$t_{3n}=6s_{2n}+t_{2n},\qquad 0\leq t_{2n}\leq 5,$$ $$t_{2n}=3s_{1n}+t_{1n},\qquad 0\leq t_{1n}\leq 2.$$

Set $S=\{12,20+15k,23+15k,27+15k\mid k\geq0\}$ and define

$$\nu(n)= \begin{cases} 2t_{1n}+3s_{1n}+4s_{2n}+5s_{3n}+6s_{4n},&n\notin S,\\ 2t_{1n}+3s_{1n}+4s_{2n}+5s_{3n}+6s_{4n}-1,&n\in S\setminus\{20+15k\mid k\geq0\},\\ 2t_{1n}+3s_{1n}+4s_{2n}+5s_{3n}+6s_{4n}-3,&\text{otherwise.} \end{cases}$$

Explicit Apéry-set conjecture for $\Gamma_6$. The Apéry set of $\Gamma_6$ with respect to $a$ is

$$\operatorname{Ap}(\Gamma_6,a)=\{\nu(n)a+nd\mid n\in\{1,\ldots,a-1\}\}\cup\{0\}.$$

This gives an explicit description of the Apéry set in the six-generator case and is intended to support the study of unique expansions and tangent cones; the source gives no resolution of the formula.