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).
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." Tetrahedron24, 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. Math19,
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.
-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),
-polyhexes
with holes for 
,
7, 8, ... are 1, 2, 13, 67, 404, ... (OEIS A038144;
Myers), the first few of which are illustrated above.
-hexagon one-sided polyhexes are 1, 1, 3, 10, 33, 147, 620,
2821, 12942, 60639, 286190, 1364621, 6545430, ... (OEIS A006535).
