OFFSET
1,6
COMMENTS
A cell of a polyiamond is saturated when all three of its edge-neighbors are present, so the cell is completely surrounded by the polyiamond.
a(n) counts the free polyiamonds (distinct up to the rotations and reflections of the triangular lattice) on n cells with at least one saturated cell.
The smallest polyiamond with a saturated cell has four cells -- a cell with its three neighbors -- so a(1) = a(2) = a(3) = 0. The sequence is a subcount of A000577 (free polyiamonds), so a(n) < A000577(n) for all n, because the straight strip of n triangles has no saturated cell. The proportion a(n)/A000577(n) increases with n, reaching 0.9955 at n = 17.
The free polyiamonds with no saturated cell (the complement A000577(n) - a(n)) are exactly those in which every cell has at most two edge-neighbors, so their cells form a single path or cycle. This complement is small, because almost all polyiamonds are saturated.
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 polyiamonds, so all but an exponentially small fraction of n-cell polyiamonds contain one; hence a(n)/A000577(n) tends to 1 and almost all free polyiamonds are saturated, with a(n) sharing the growth rate of A000577(n).
LINKS
Peter Exley, Paper and figures, GitHub.
Neal Madras, A pattern theorem for lattice clusters, Annals of Combinatorics 3 (1999), 357-384; arXiv:math/9902161 [math.PR], 1999.
FORMULA
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Peter Exley, Jun 15 2026
EXTENSIONS
a(18)-a(29) from Pontus von Brömssen, Jun 17 2026
a(30) (using formula) from Pontus von Brömssen, Jun 27 2026
STATUS
approved