OFFSET
1,1
COMMENTS
An n-cell connected hole H on the trihexagonal tiling is a connected set of n cells (each cell either an equilateral triangle or a regular hexagon in the 3.6.3.6 vertex figure). A connected enclosing shell is a connected set S of cells, disjoint from H, such that every cell edge-adjacent to H is in S, and the complement of H union S is connected (so the exterior is a single region; S has no internal cavities). a(n) is the minimum size of such a shell over all connected n-cell holes.
The trihexagonal tiling is also called the kagome lattice. It has equilateral triangles and regular hexagons in a 2:1 ratio; the tiling is edge-transitive, with every edge separating one triangle from one hexagon, so two triangles never share an edge. Symmetry group p6m.
From Peter Exley, Jun 06 2026: (Start)
On this tiling two hexagons never share an edge, so the corona of a compact hole is a ring of isolated hexagons; the shell adds one bridging triangle between consecutive corona hexagons, giving a(n) = 2*c - 1 with c the corona size. The shell grows like the perimeter of an area-n hole.
Conjecture (UNVERIFIED, matches a(1..22)): a(n) grows as sqrt(18*n); equivalently the largest hole an s-cell shell can enclose is floor((s^2 + 6)/18), and the achievable shell sizes are 5, 11, and the odd integers >= 15 (so 7, 9, and 13 never occur). Each value is established as minimal over holes lying within a bounded region of the 3.6.3.6 tiling; that the region is large enough to exclude a smaller shell arising from a larger hole is supported by a wider-region check but not proved, so global minimality is conjectured. (End)
LINKS
Peter Exley, Paper and figures, GitHub.
EXAMPLE
For n = 1, a(1) = 5: hole is one triangle; corona of 3 hexagons in 3 components, 2 bridges.
For n = 2, a(2) = 11: corona of 7 cells in 5 components, 4 bridges.
For n = 3, a(3) = 11: corona of 8 cells in 4 components, 3 bridges; same shell size as a(2).
For n = 7, a(7) = 11: end of the first plateau; corona of 6 cells in 6 components, 5 bridges. A larger hole need not require a larger shell.
For n = 8, a(8) = 15: first step beyond the plateau; corona of 10 cells in 6 components, 5 bridges.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Peter Exley, May 07 2026
EXTENSIONS
a(11)-a(22) from Peter Exley, Jun 06 2026
STATUS
approved