cp's OEIS Frontend

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

%I A000228 M2682 N1072 #106 Feb 16 2025 08:32:20
%S A000228 1,1,3,7,22,82,333,1448,6572,30490,143552,683101,3274826,15796897,
%T A000228 76581875,372868101,1822236628,8934910362,43939164263,216651036012,
%U A000228 1070793308942,5303855973849,26323064063884,130878392115834,651812979669234,3251215493161062,16240020734253127,81227147768301723,406770970805865187,2039375198751047333
%N A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.
%C A000228 From _Markus Voege_, Nov 24 2009: (Start)
%C A000228 On the difference between this sequence and A038147:
%C A000228 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.
%C A000228 If I recall correctly, polyhexes are clusters of regular hexagons that are joined at the edges and are LOCALLY embeddable in the hexagonal lattice.
%C A000228 "Planar polyhexes" are polyhexes that are GLOBALLY embeddable in the honeycomb lattice.
%C A000228 Example: (Planar) polyhex with 6 cells (x) and a hole (O):
%C A000228 .. x x
%C A000228 . x O x
%C A000228 .. x x
%C A000228 Polyhex with 6 cells that is cut open (I):
%C A000228 .. xIx
%C A000228 . x O x
%C A000228 .. x x
%C A000228 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.
%C A000228 Illegal configuration with cut (I):
%C A000228 .. xIx
%C A000228 . x x x
%C A000228 .. x x
%C A000228 This configuration is NOT a polyhex since the vertex at
%C A000228 .. xIx
%C A000228 ... x
%C A000228 is not embeddable in the honeycomb lattice.
%C A000228 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.
%C A000228 The (planar) polyhexes are "free" configurations, in contrast to "fixed" configurations as in A001207 = Number of fixed hexagonal polyominoes with n cells.
%C A000228 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.
%C A000228 The configuration
%C A000228 . x x .... x
%C A000228 .. x .... x x
%C A000228 is counted once as free and twice as fixed configurations.
%C A000228 Since most configurations have no symmetry, (A001207 / A000228) -> 12 for n -> infinity. (End)
%D A000228 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.
%D A000228 A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, 3599-3609
%D A000228 M. Gardner, Polyhexes and Polyaboloes. Ch. 11 in Mathematical Magic Show. New York: Vintage, pp. 146-159, 1978.
%D A000228 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.
%D A000228 J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
%D A000228 W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
%D A000228 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000228 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000228 John Mason and Robert A. Russell, <a href="/A000228/b000228.txt">Table of n, a(n) for n = 1..36</a>
%H A000228 Frédéric Chyzak, Ivan Gutman, and Peter Paule, <a href="https://match.pmf.kg.ac.rs/electronic_versions/Match40/match40_139-151.pdf">Predicting the number of hexagonal systems with 24 and 25 hexagons</a>, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
%H A000228 A. Clarke, <a href="http://www.recmath.com/PolyPages/PolyPages/Polyhexes.html">Polyhexes</a>
%H A000228 F. Harary and R. C. Read, <a href="https://doi.org/10.1017/S0013091500009135">The enumeration of tree-like polyhexes</a>, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
%H A000228 D. Gouyou-Beauchamps and P. Leroux, <a href="https://arxiv.org/abs/math/0403168">Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice</a>, arXiv:math/0403168 [math.CO], 2004.
%H A000228 M. Keller, <a href="http://www.solitairelaboratory.com/polyenum.html">Counting polyforms</a>
%H A000228 D. A. Klarner, <a href="http://dx.doi.org/10.4153/CJM-1967-080-4">Cell growth problems</a>, Canad. J. Math. 19 (1967) 851-863.
%H A000228 J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match16/match16_119-134.pdf">On the total number of polyhexes</a>, Match, No. 16 (1984), 119-134.
%H A000228 Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, <a href="https://doi.org/10.1007/s00026-022-00631-1">Extremal {p, q}-Animals</a>, Ann. Comb. (2023), p. 3.
%H A000228 John Mason, <a href="/A000228/a000228_2.pdf">Counting polyhexes of size 36</a>, updated Oct 27 2023.
%H A000228 Joseph Myers, <a href="http://www.polyomino.org.uk/mathematics/polyform-tiling/">Polyomino, polyhex and polyiamond tiling</a>
%H A000228 Ed Pegg, Jr., <a href="http://demonstrations.wolfram.com/PolyformExplorer/">Illustrations of polyforms</a>
%H A000228 Jaime Rangel-Mondragon, <a href="https://web.archive.org/web/20190411024906/http://www.mathematica-journal.com/issue/v9i3/polyominoes.html">Polyominoes and Related Families</a>, The Mathematica Journal, 9:3 (2005), 609-640.
%H A000228 N. J. A. Sloane, <a href="/A000228/a000228.gif">Illustration of initial terms</a>
%H A000228 N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, <a href="https://doi.org/10.1351/pac198855020379">Computer Generation of Isomeric Structures</a>, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
%H A000228 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyhex.html">Polyhex</a>.
%Y A000228 Equals (A006535 + A030225)/2.
%Y A000228 Cf. A036359, A002216, A005963, A000228, A001998, A018190.
%Y A000228 Cf. A001207, A057973, A364306.
%Y A000228 Cf. also A000105, A000577, A103465, A057779, A258206, A258019.
%K A000228 nonn,nice,hard
%O A000228 1,3
%A A000228 _N. J. A. Sloane_
%E A000228 a(13) from _Achim Flammenkamp_, Feb 15 1999
%E A000228 a(14) from Brendan Owen, Dec 31 2001
%E A000228 a(15) from _Joseph Myers_, May 05 2002
%E A000228 a(16)-a(20) from _Joseph Myers_, Sep 21 2002
%E A000228 a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
%E A000228 a(22)-a(30) from _John Mason_, Jul 18 2023