Non-adjacent triples of cells in a triangular grid: proofs of three conjectured generating functions
Abstract
Let Tn be the triangular grid of side n, with its n2 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 C3 and under the full symmetry group D3 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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BibTeX
@misc{niu2026nonadjacent,
title = {{Non-adjacent triples of cells in a triangular grid: proofs of three conjectured generating functions}},
author = {Tong Niu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33023},
eprint = {2610.33023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33023},
}
RIS
TY - UNPB
TI - Non-adjacent triples of cells in a triangular grid: proofs of three conjectured generating functions
AU - Tong Niu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.33023
UR - https://arxiv.org/abs/2610.33023
AB - 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.
ER -