An analog of the polyominoes and polyiamonds in which collections of regular hexagons are arranged with adjacent sides. They are
also called hexes, hexas, or polyfrobs (Beeler 1972). For the 4-hexes (tetrahexes),
the possible arrangements are known as the bee, bar,
pistol, propeller, worm,
arch, and wave.
The numbers of geometrically planar -polyhexes for , 2, ... are 1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490,
143552, ... (OEIS A000228; Klarner 1967, Balaban
and Harary 1968, Harary and Read 1970, Lunnon 1972, Gardner 1978, Knop et al.
1984, Gardner 1988),
The numbers of -polyhexes
with holes for ,
7, 8, ... are 1, 2, 13, 67, 404, ... (OEIS A038144;
Myers), the first few of which are illustrated above.
"One-sided" polyhexes are considered to be fixed in the plane, and so mirror images are counted separately. The numbers of -hexagon one-sided polyhexes are 1, 1, 3, 10, 33, 147, 620,
2821, 12942, 60639, 286190, 1364621, 6545430, ... (OEIS A006535).
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