arXiv:2610.33023v1 Announce Type: new Abstract: Let $\Tn$ be the triangular grid of side $n$, with its $n^2$ unit cells, and let two cells be side-adjacent if they share an edge and point-adjacent if they share a vertex. We count the orbits of $3$-element sets of pairwise non-adjacent cells under the rotation group $C_3$ and under the full symmetry group $D_3$ of $\Tn$. Three of the four resulting sequences are OEIS A397730, A397731 and A397732, for each of which the only formula on record is a generating function conjectured by Sean A.\ Irvine on 14 July 2026. We prove all three conjectures. The route is Burnside's lemma, together with exact lattice counts for each fixed-point term. The side-adjacency graph turns out to be bipartite, so its triangle count vanishes, and the triangles of the point-adjacency graph have a clean structural description, which gives that count too. Along the way we obtain the first explicit closed forms for the three sequences, each a quasi-polynomial of degree $6$ and period $3$, $6$, $3$ respectively. The same machinery settles the fourth member of the family, whose closed form has been recorded at A289230 without proof since 2017, and the labeled count A289224 that feeds it.
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