A038144 Number of planar n-hexes, or polyhexes (in the sense of A000228, so rotations and reflections count as the same shape) with at least one hole.

0, 0, 0, 0, 0, 1, 2, 13, 67, 404, 2323, 13517, 76570, 429320, 2373965, 13004323, 70641985, 381260615, 2046521491, 10936624026, 58228136539, 309048278652, 1635939163344, 8640183332631, 45543853593056, 239662654145342, 1259297620368388, 6608341442275135

Offset

1,7

Comments

Apparently the same as the number of hexagonal planar circulenes (planar rings of hexagons) with n cells, although the two sequences may in fact differ for large n.

From John Mason, Sep 03 2025: (Start)

The number of holes in a polyhex is given by the formula: H = n + 1 - (3i + j + 2k)/6, where:

n is the size (area) of the polyhex;

i is the number of vertices common to 3 hexagons;

j is the number of vertices common to 2 hexagons;

k is the number of vertices common to 1 hexagon. (End)

References

J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.

Links

Table of n, a(n) for n=1..28.

Craig Knecht, Tiling of a fixed pattern of holes in a frame.

J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.

Joseph Myers, Polyomino, polyhex and polyiamond tiling.

Formula

a(n) = A000228(n) - A018190(n). - John Mason, Sep 28 2025

Crossrefs

Cf. A018190, A038143, A038147, A000228, A018190.

Sequence in context: A045764 A106999 A054566 * A097977 A373199 A136780

Adjacent sequences: A038141 A038142 A038143 * A038145 A038146 A038147

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Joseph Myers, May 05 2002

Further terms from Joseph Myers, Nov 06 2003

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

Link edited by Joseph Myers, Nov 17 2010

a(22)-a(28) from John Mason, Sep 28 2025

Status

approved