OFFSET
1,8
COMMENTS
A cell of a polyhex is saturated when all six of its edge-neighbors are present, so the cell is completely surrounded by the polyhex.
a(n) counts the free polyhexes (distinct up to the hexagon symmetry group of order 12) of n cells with at least one saturated cell.
The smallest polyhex with a saturated cell is the seven-cell flower, a hexagon with its six neighbors, so a(1) through a(6) are 0. The sequence is a subcount of A000228 (free polyhexes), so a(n) < A000228(n) for all n, because the straight strip of n cells has no saturated cell.
The proportion a(n)/A000228(n) rises with n but very slowly, reaching about 0.003 at n = 12. A saturated cell needs six present neighbors, more than on the square or triangular lattices, so interior cells appear later here.
The complementary count A000228(n) - a(n) = 1, 1, 3, 7, 22, 82, 332, 1446, 6560, 30428, 143195, 681087 (for n = 1..12) is the number of free polyhexes with no saturated cell, in which every cell keeps at least one exposed edge (at most five of its six edge-neighbors present). These shapes remain the large majority over the computed range.
By N. Madras's pattern theorem for lattice clusters (Annals of Combinatorics 3 (1999), 357-384), a saturated cell is a local pattern occurring in the interior of large polyhexes, so all but an exponentially small fraction of n-cell polyhexes contain one; hence a(n)/A000228(n) tends to 1 and almost all free polyhexes are saturated, with a(n) sharing the growth rate of A000228(n). The approach is the slowest of the three regular lattices.
REFERENCES
N. Madras, A pattern theorem for lattice clusters, Annals of Combinatorics 3 (1999), 357-384.
LINKS
Peter Exley, Paper and figures, GitHub.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Peter Exley, Jun 16 2026
STATUS
approved