A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, 3274826, 15796897, 76581875, 372868101, 1822236628, 8934910362, 43939164263, 216651036012, 1070793308942, 5303855973849, 26323064063884, 130878392115834, 651812979669234, 3251215493161062, 16240020734253127, 81227147768301723, 406770970805865187, 2039375198751047333

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

Comments

From Markus Voege, Nov 24 2009: (Start)

On the difference between this sequence and A038147:

The first term that differs is for n=6; for all subsequent terms, the number of polyhexes is larger than the number of planar polyhexes.

If I recall correctly, polyhexes are clusters of regular hexagons that are joined at the edges and are LOCALLY embeddable in the hexagonal lattice.

"Planar polyhexes" are polyhexes that are GLOBALLY embeddable in the honeycomb lattice.

Example: (Planar) polyhex with 6 cells (x) and a hole (O):

.. x x

. x O x

.. x x

Polyhex with 6 cells that is cut open (I):

.. xIx

. x O x

.. x x

This polyhex is not globally embeddable in the honeycomb lattice, since adjacent cells of the lattice must be joined. But it can be embedded locally everywhere. It is a start of a spiral. For n>6 the spiral can be continued so that the cells overlap.

Illegal configuration with cut (I):

.. xIx

. x x x

.. x x

This configuration is NOT a polyhex since the vertex at

.. xIx

... x

is not embeddable in the honeycomb lattice.

One has to keep in mind that these definitions are inspired by chemistry. Hence, potential molecules are often the motivation for these definitions. Think of benzene rings that are fused at a C-C bond.

The (planar) polyhexes are "free" configurations, in contrast to "fixed" configurations as in A001207 = Number of fixed hexagonal polyominoes with n cells.

A000228 (planar polyhexes) and A001207 (fixed hexagonal polyominoes) differ only by the attribute "free" vs. "fixed," that is, whether the different orientations and reflections of an embedding in the lattice are counted.

The configuration

. x x .... x

.. x .... x x

is counted once as free and twice as fixed configurations.

Since most configurations have no symmetry, (A001207 / A000228) -> 12 for n -> infinity. (End)

References

• A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2516.
• A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, 3599-3609
• M. Gardner, Polyhexes and Polyaboloes. Ch. 11 in Mathematical Magic Show. New York: Vintage, pp. 146-159, 1978.
• M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.
• J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
• W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• John Mason and Robert A. Russell, Table of n, a(n) for n = 1..36
• Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
• A. Clarke, Polyhexes
• F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
• D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
• M. Keller, Counting polyforms
• D. A. Klarner, Cell growth problems, Canad. J. Math. 19 (1967) 851-863.
• J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
• Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
• John Mason, Counting polyhexes of size 36, updated Oct 27 2023.
• Joseph Myers, Polyomino, polyhex and polyiamond tiling
• Ed Pegg, Jr., Illustrations of polyforms
• Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
• N. J. A. Sloane, Illustration of initial terms
• N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
• Eric Weisstein's World of Mathematics, Polyhex.

Crossrefs

Equals (A006535 + A030225)/2.

Cf. A036359, A002216, A005963, A000228, A001998, A018190.

Cf. A001207, A057973, A364306.

Cf. also A000105, A000577, A103465, A057779, A258206, A258019.

Extensions

a(13) from Achim Flammenkamp, Feb 15 1999

a(14) from Brendan Owen, Dec 31 2001

a(15) from Joseph Myers, May 05 2002

a(16)-a(20) from Joseph Myers, Sep 21 2002

a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

a(22)-a(30) from John Mason, Jul 18 2023