Polyhex
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).
A simple connected polyhex is called a fusene.
SEE ALSO: Benzenoid,
Fullerene,
Fusene,
Polyform,
Polyhex
Tiling,
Polyiamond,
Polyomino,
Polyplet
REFERENCES:
Balaban, A. T. "Enumeration of Cyclic Graphs." In Chemical Applications of Graph Theory (Ed. A. T. Balaban). London: Academic
Press, pp. 63-105, 1976.
Balaban, A. T. and Harary, F. "Chemical Graphs V: Enumeration and Proposed Nomenclature of Benzenoid Cata-Condensed Polycyclic Aromatic Hydrocarbons."
Tetrahedron 24, 2505-2506, 1968.
Balasubramanian, K.; Kauffman, J. J.; Koski, W. S.; and Balaban, A. T. "Graph Theoretical Characterization and Computer Generation of Certain Carcinogenic
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.recmath.com/PolyPages/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.
Harary, F. Graph
Theory. Reading, MA: Addison-Wesley, pp. 178-197, 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.
Referenced on Wolfram|Alpha:
Polyhex
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Weisstein, Eric W. "Polyhex." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Polyhex.html
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