OFFSET
1,1
COMMENTS
The 4.8.8 (truncated square) tiling has regular squares and octagons in a 1:1 ratio, symmetry group p4mm. For a connected n-cell hole H, the corona C(H) is the set of cells edge-adjacent to H but not in H. An enclosing shell S is a connected superset of C(H), disjoint from H, with the cells outside H union S also connected. a(n) is the minimum |S|.
Conjectured (UNVERIFIED beyond n = 23): the minimum shell equals the corona of the optimal hole, |S| = |C(H)|; and a(n) = 2*ceiling((n + 17)/5) for n >= 6, equivalently a(n) = a(n - 5) + 2 for n >= 11. - Peter Exley, Jun 05 2026
LINKS
Peter Exley, Paper and figures, GitHub.
EXAMPLE
For n = 1, a(1) = 4: a single square cell as the hole has corona of size 4 (its four octagon neighbors), and that corona is connected and forms a valid enclosing shell.
For n = 2, a(2) = 8: every 2-cell connected hole on 4.8.8 contains at least one octagon; the corona of a square plus an adjacent octagon has 8 cells which form a connected ring around the hole.
For n = 5, a(5) = 8: the last n in the first plateau (a(n) = 8 for n = 2..5); adding interior cells to the hole does not enlarge the corona while the optimal shape stays compact.
CROSSREFS
Cf. A182619 (analogous shell sequence on the regular hexagonal grid), A227004 (coordination sequence for the truncated-square 4.8.8 tiling; different "shell" concept, concentric coordination shells, included as disambiguation), A283056 (smallest polyomino admitting a hole of size n on the regular square grid).
KEYWORD
nonn,more,changed
AUTHOR
Peter Exley, May 05 2026
EXTENSIONS
a(21)-a(23) from Peter Exley, Jun 05 2026
STATUS
approved