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,
... (Sloane's 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, ... (Sloane's 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,
... (Sloane's A006535).
A simple connected polyhex is called a fusene.
Balaban, A. T. "Enumeration of Cyclic Graphs." In Chemical Applications of Graph Theory (Ed. A. T. Balaban).
London: Academic Press, pp. 63-105, 1976.
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Benzenoid Hydrocarbons and Identification." J. Comput. Chem. 1,
149-157, 1980.
Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48-50,
Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/polyominos.html#item112.
Beineke, L. W. and Pippert, R. E. "On the Enumeration of Planar Trees
of Hexagons." Glasgow Math. J. 15, 131-147, 1974.
Brinkmann, G.; Caporossi, G.; and Hansen, P. "A Constructive Enumeration of
Fusenes and Benzenoids." J. Algorithms. 45, 155-166, 2002.
Brinkmann, G.; Caporossi, G.; and Hansen, P. "A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons." J. Chem. Inf. Comput. Sci. 43,
842-851, 2003.
Clarke, A. L. "Polyhexes." http://www.geocities.com/alclarke0/PolyPages/Polyhexes.html.
Cyvin, S. J.; Brunvoll, J.; Xiaofeng, G.; and Fuji, Z. "Number of Perifusenes
with One Internal Vertex." Rev. Roumaine Chem. 38, 65-77, 1993.
Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydrocarbons. 1. Isomer Enumeration of Fused Polycyclic Aromatic Hydrocarbon." J. Chem. Inf.
Comput. Sci. 22, 15-22, 1982.
Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydrocarbons. 2. Polycyclic Aromatic Hydrocarbons Containing Tetragonal, Pentagonal, Heptagonal, and
Octagonal Rings." J. Chem. Inf. Comput. Sci. 22, 139-152, 1982.
Dias, J. R. "A Periodic Table for Polycyclic Aromatic Hydrocarbons. 3. Enumeration of All the Polycyclic Conjugated Isomers of Pyrene Having Ring Sizes
Ranging from 3 to 9." Math. Chem (Mülheim/Ruhr) 14, 83-138,
1983.
Gardner, M. "Polyhexes and Polyaboloes." Ch. 11 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions
and Other Mathematical Sleight-of-Mind from Scientific American. New York:
Vintage, pp. 146-159, 1978.
Gardner, M. "Tiling with Polyominoes, Polyiamonds, and Polyhexes." Ch. 14 in Time Travel and Other Mathematical Bewilderments. New York:
W. H. Freeman, pp. 175-187, 1988.
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed.
Princeton, NJ: Princeton University Press, pp. 92-93, 1994.
Harary, F. "Graphical Enumeration Problems." In Graph Theory and Theoretical Physics (Ed. F. Harary).
London: Academic Press, pp. 1-41, 1967.
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1994.
Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, 1973.
Harary, F. and Read, R. C. "The Enumeration of Tree-Like Polyhexes."
Proc. Edinburgh Math. Soc. 17, 1-13, 1970.
Keller, M. "Counting Polyforms." http://members.aol.com/wgreview/polyenum.html.
Klarner, D. A. "Cell Growth Problems." In Canad. J. Math 19,
851-863, 1967.
Knop, J. V.; Szymanski, K.; Jeričević, Ž.; and Trinajstić, N. "On the Total Number of Polyhexes." Match: Commun. Math. Chem.,
No. 16, 119-134, Aug. 1984.
Lunnon, W. F. "Counting Hexagonal and Triangular Polyominoes." In Graph Theory and Computing (Ed. R. C. Read).
New York: Academic Press, pp. 87-100, 1972.
Myers, J. "Polyomino Tiling." http://www.srcf.ucam.org/~jsm28/tiling/.
Palmer, E. M. "Variations of the Cell Growth Problem." In Graph Theory and Applications: Proceedings of the Conference at
Western Michigan University, Kalamazoo, Mich., May 10-13, 1972 (Ed. Y. Alavi,
D. R. Lick, and A. T. White). New York: Springer-Verlag, pp. 214-223,
1972.
Sloane, N. J. A. Sequences A000228/M2682, A038144, and A006535/M2846 in "The On-Line Encyclopedia of Integer
Sequences."
Vichera, M. "Polyforms." http://www.vicher.cz/puzzle/polyforms.htm.
von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 342-343, 1993.
Weisstein, E. W. "Books about Polyominoes." http://www.ericweisstein.com/encyclopedias/books/Polyominoes.html.
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