A038147 -id:A038147 - cp's OEIS frontend

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

N. J. A. Sloane

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)

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).

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.

Equals (A006535 + A030225)/2.
Cf. A036359, A002216, A005963, A000228, A001998, A018190.
Cf. A001207, A057973, A364306.
Cf. also A000105, A000577, A103465, A057779, A258206, A258019.

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

A018190 Number of planar simply-connected polyhexes (or benzenoid hydrocarbons) with n hexagons.

Original entry on oeis.org

1, 1, 3, 7, 22, 81, 331, 1435, 6505, 30086, 141229, 669584, 3198256, 15367577, 74207910, 359863778, 1751594643, 8553649747, 41892642772, 205714411986, 1012565172403, 4994807695197, 24687124900540, 122238208783203

Offset: 1

N. J. A. Sloane, Viviane Rochon (viviane(AT)crt.umontreal.ca), Gilles Caporossi (gillesc(AT)crt.umontreal.ca)

J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298.

N. J. A. Sloane, Table of n, a(n) for n = 1..35 [from Vöge et al.]
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A constructive enumeration of fusenes and benzenoids, Journal of Algorithms 45 (2002), pp. 155-166.
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons, J. Chem. Inf. Comput. Sci., vol. 43 (2003) 842-851.
G. Caporossi, P. Hansen, Enumeration of Polyhex Hydrocarbons to h = 21, J. Chem. inf. Comput. Sci. 38 (4) (1998) 610-619, Table 1.
James Chapman, Judith Foos, Andrew Nelson, Elizabeth J. Hartung, and Aaron Williams, Pairwise disagreements of Kekulé, Clar, and Fries numbers for benzenoids: a mathematical and computational investigation, arXiv:1804.06071 [math.PR], 2018.
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.
J. L. Faulon, D. Visco, Jr., and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
J. V. Knop, W. R. Müller, K. Szymanski, and N. Trinajstić, Use of small computers for large computations: enumeration of polyhex hydrocarbons, J. Chem. Inf. Comput. Sci., 30 (1990), 159-160.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Elena V. Konstantinova and Maxim V. Vidyuk, Discriminating tests of information and topological indices. Animals and trees, J. Chem. Inf. Comput. Sci. 43 (2003), 1860-1871.
Lucia Moura and Ivan Stojmenovic, Backtracking and Isomorph-Free Generation of Polyhexes, Table 2.1 on p. 50 of Handbook of Applied Algorithms (2008). Note a(8) is given as 1453!
S. Nikolić, N. Trinajstić, J. V. Knop, W. R. Müller, On the concept of the weighted spanning tree of dualist, J. Math. Chem. 4 (1990), 357-375.
R. Tošić, D. Mašulović, I. Stojmenović, J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., 1995, 35, 181-187.
N. Trinajstić, Z. Jerievi, J. V. Knop, W. R. Müller, and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Markus Vöge, Anthony J. Guttmann, and Iwan Jensen, On the Number of Benzenoid Hydrocarbons, Journal of Chemical Information and Computer Sciences, 42(3) (2002), 456-466.
Eric Weisstein's World of Mathematics, Polyhex.
Eric Weisstein's World of Mathematics, Benzenoid
Eric Weisstein's World of Mathematics, Fusene [That page refers to this sequence as "[simply-connected] catafusenes", but in fact the polyhexes counted by this sequence are not necessarily catacondensed (i.e. tree-like). - _Andrey Zabolotskiy_, Apr 11 2025]

Cf. A000228, A002216, A018190, A038142-A038147.

More terms from Joseph Myers, Nov 06 2003

Further terms added by N. J. A. Sloane from Brinkmann et al. (2003). Jun 04 2005

A323930 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

Original entry on oeis.org

1, 1, 3, 7, 22, 81, 331, 1436, 6510, 30129, 141512, 671538, 3210620, 15443871, 74662005, 362506902

Offset: 1

N. J. A. Sloane, Feb 09 2019

N. Trinajstić, S. Nikolić, J. V. Knop, W. R. Müller and K. Szymanski, Computational Chemical Graph Theory: Characterization, Enumeration, and Generation of Chemical Structures by Computer Methods, Ellis Horwood, 1991. [incorrectly gives a(12) = 671512 in Table 4.13]

Björg N. Cyvin, Jon Brunvoll and Sven J. Cyvin, Enumeration of benzenoid systems and other polyhexes, p. 65-180 in: I. Gutman (ed.), Advances in the Theory of Benzenoid Hydrocarbons II, Springer, 1992.
Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron 42.19 (1986): 5291-5299. See Table 1.
Lorentz Jäntschi, Topological Characterization of a Complete Set of Small‐Sized Graphene Sheets Using Molecular Descriptors With Energy Storage Applications, Ener. Stor. (2025) Vol. 7, Issue 6, Art No. e70253.

Cf. A003104, A323923-A323929.
A018190 is a very similar sequence with (as He and He remark) a slightly different definition. Cf. A000228, A038147.
Cf. A038140, A038141.

a(n) = A018190(n) + A038140(n) + A038141(n). - Andrey Zabolotskiy, Feb 16 2023

a(10)-a(16) from Cyvin, Brunvoll & Cyvin (Table 1) added by Andrey Zabolotskiy, Feb 08 2023

A038144 Number of planar n-hexes, or polyhexes (in the sense of A000228, so rotations and reflections count as the same shape) with at least one hole.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 13, 67, 404, 2323, 13517, 76570, 429320, 2373965, 13004323, 70641985, 381260615, 2046521491, 10936624026, 58228136539

Offset: 1

N. J. A. Sloane

nonn

Apparently the same as the number of hexagonal planar circulenes (planar rings of hexagons) with n cells, although the two sequences may in fact differ for large n.

J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.

Craig Knecht, Tiling of a fixed pattern of holes in a frame.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Joseph Myers, Polyomino, polyhex and polyiamond tiling

Cf. A018190, A038143, A038147, A000228, A018190.

More terms from Joseph Myers, May 05 2002
Further terms from Joseph Myers, Nov 06 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
Link edited by Joseph Myers, Nov 17 2010

A036496 Number of lines that intersect the first n points on a spiral on a triangular lattice. The spiral starts at (0,0), goes to (1,0) and (1/2, sqrt(3)/2) and continues counterclockwise.

Original entry on oeis.org

0, 3, 5, 6, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29, 29, 29, 30

Offset: 0

Mario VELUCCHI (mathchess(AT)velucchi.it)

The triangular lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called a hexagonal lattice.
Conjecture: a(n) is half the minimal perimeter of a polyhex of n hexagons. - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 06 2002. This conjecture follows from the Brunvoll et al. reference. - Sascha Kurz, Mar 17 2008
From a spiral of n triangular lattice points, we can get a polyhex of n hexagons with min perimeter by replacing each point on the spiral by a hexagon. - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002

			For n=3 the 3 points are (0,0), (1,0), (1/2, sqrt(3)/2) and there are 3 lines: 2 horizontal, 2 sloping at 60 degrees and 2 at 120 degrees, so a(3)=6.

J. Bornhoft, G. Brinkmann, J. Greinus, Pentagon-hexagon-patches with short boundaries, European J. Combin. 24 (2003), 517-529.
F. Harary and H. Harborth, Extremal animals, Journal of Combinatorics, Information, & System Sciences, Vol. 1, 1-8, (1976).
W. C. Yang, Maximal and minimal polyhexes, manuscript, 2002.
W. C. Yang, PhD thesis, Computer Sciences Department, University of Wisconsin-Madison, 2003.
J. Brunvoll, B.N. Cyvin and S.J Cyvin, More about extremal animals, Journal of Mathematical Chemistry Vol. 12 (1993), pp. 109-119

Harvey P. Dale, Table of n, a(n) for n = 0..1000
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A001399, A038147.

Join[{0},Ceiling[Sqrt[12*Range[80]-3]]] (* Harvey P. Dale, May 26 2017 *)

If n >= 1, a(n) = ceiling(sqrt(12n - 3)). - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 06 2002

More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000

cp's OEIS Frontend

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

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A018190 Number of planar simply-connected polyhexes (or benzenoid hydrocarbons) with n hexagons.

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A323930 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

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A038144 Number of planar n-hexes, or polyhexes (in the sense of A000228, so rotations and reflections count as the same shape) with at least one hole.

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A036496 Number of lines that intersect the first n points on a spiral on a triangular lattice. The spiral starts at (0,0), goes to (1,0) and (1/2, sqrt(3)/2) and continues counterclockwise.

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A038146 Number of n-celled helicenes with peri-fragments.

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A038145 Number of n-celled helicenes without holes.

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A176638 Partial sums of A018190.

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