A018190 -id:A018190 - cp's OEIS frontend

A176638 Partial sums of A018190.

1, 2, 5, 12, 34, 115, 446, 1881, 8386, 38472, 179701, 849285, 4047541, 19415118, 93623028, 453486806, 2205081449, 10758731196, 52651373968, 258365785954, 1270930958357, 6265738653554, 30952863554094, 153191072337297

Offset: 1

Jonathan Vos Post, Apr 22 2010

nonn

Partial sums of number of planar simply-connected polyhexes (or benzenoid hydrocarbons) with n hexagons. The only known primes in the partial sum are 2 and 5.

			a(6) = 1 + 1 + 3 + 7 + 22 + 81 = 115.

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

a(n) = SUM[i=1..n] A018190(i).

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

Original entry on oeis.org

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

A000104 Number of n-celled free polyominoes without holes.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 35, 107, 363, 1248, 4460, 16094, 58937, 217117, 805475, 3001127, 11230003, 42161529, 158781106, 599563893, 2269506062, 8609442688, 32725637373, 124621833354, 475368834568, 1816103345752, 6948228104703, 26618671505989, 102102788362303

Offset: 0

N. J. A. Sloane

J. S. Madachy, Pentominoes - Some Solved and Unsolved Problems, J. Rec. Math., 2 (1969), 181-188.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
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, Table of n, a(n) for n = 0..40
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.
John Mason, Description of counting programs
John Mason, Programs for calculation of numbers of unholey polyominoes
Lucia Moura and Ivan Stojmenovic, Backtracking and Isomorph-Free Generation of Polyhexes, Table 2.2 on p. 55 of Handbook of Applied Algorithms (2008).
W. R. Muller, K. Szymanski, J. V. Knop, and N. Trinajstic, On the number of square-cell configurations, Theor. Chim. Acta 86 (1993) 269-278
Tomás Oliveira e Silva, Enumeration of polyominoes
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy)
R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Cf. A000105, row sums of A308300, A006746, A056877, A006748, A056878, A006747, A006749, A054361, A070765 (polyiamonds), A018190 (polyhexes), A266549 (by perimeter).

Cf. A056879, A056880, A056881, A056882, A006724, A357647, A357648.

a(n) = A000105(n) - A001419(n). - John Mason, Sep 06 2022

a(n) = (4*A056879(n) + 4*A056881(n) + 4*A056883(n) + 6*A056880(n) + 6*A056882(n) + 6*A357647(n) + 7*A357648(n) + A006724(n)) / 8. - John Mason, Oct 10 2022

Extended to n=26 by Tomás Oliveira e Silva

a(27)-a(28) from Tomás Oliveira e Silva's page added by Andrey Zabolotskiy, Oct 02 2022

A258206 Number of strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotations and turning over.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 312, 744, 2291, 6186, 18714, 53793, 162565, 482416, 1467094, 4436536, 13594266, 41640513, 128564463, 397590126, 1236177615, 3852339237, 12053032356, 37802482958, 118936687722, 375079338476

Offset: 1

Antti Karttunen, May 31 2015

Differs from A057779 for the first time at n=12 as here a(12) = 97, one less than A057779(12) because this sequence excludes polyhexes with holes, the smallest which contains six hexagons in a ring, enclosing a hole of one hex, having thus perimeter of 18+6 = 24 (= 2*12) edges.
Differs from A258019 for the first time at n=13 as here a(13) = 312, one less than A258019(13) because this sequence counts only strictly non-overlapping and non-touching polyhex-patterns, while A258019(13) already includes one specimen of helicene-like self-reaching structures.
If one counts these structures by the number of hexagons (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 81, ... (A018190).
a(n) is also the number of 2n-step 2-dimensional closed self-avoiding paths on honeycomb lattice, reduced for symmetry. - Luca Petrone, Jan 08 2016

S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Theory of Coronoid Hydrocarbons, Springer-Verlag, 1991. See sections 4.7 Annulene and 6.5 Annulenes.

Hugo Pfoertner, Illustration of polygons of perimeter <= 20.

Cf. A000228, A057779, A018190, A258019, A258204, A258205, A316193.

(define (A258206 n) (* (/ 1 2) (+ (A258204 n) (A258205 n))))

a(n) = (1/2) * (A258204(n) + A258205(n)).
Other observations. For all n >= 1:
a(n) <= A057779(n).
a(n) <= A258019(n).

a(14)-a(15) from Luca Petrone, Jan 08 2016
a(16)-a(23) from Cyvin, Brunvoll & Cyvin added by Andrey Zabolotskiy, Mar 01 2023
a(24)-a(32) from Bert Dobbelaere, May 12 2025

A070766 Number of polyhexes with n cells that tile the plane.

Original entry on oeis.org

1, 1, 3, 7, 22, 77, 294, 1054, 3788, 11326, 24790, 103641, 164559, 532510, 1574252, 2939898, 4761009, 21048218, 24306306, 95707819, 205176450

Offset: 1

Joseph Myers, May 05 2002

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.

Joseph Myers, Polyhex tiling

Cf. A000228-A070767, A018190-A070768, cf. A054359.

More terms from Joseph Myers, Nov 06 2003

a(20) and a(21) from Joseph Myers, Nov 17 2010

A038142 Number of planar cata-polyhexes with n cells.

Original entry on oeis.org

1, 1, 2, 5, 12, 36, 118, 411, 1489, 5572, 21115, 81121, 314075, 1224528, 4799205, 18896981, 74695032, 296275836, 1178741568, 4702507923, 18806505243, 75380203150, 302754225098, 1218239791106

Offset: 1

N. J. A. Sloane

Number of cata-condensed benzenoid hydrocarbons with n hexagons.

Planar cata-polyhexes enumerated by a(n) are the n-celled (planar) polyhexes with perimeter 4n+2, which is the maximal perimeter of an n-celled polyhex. These are such polyhexes that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge). - Tanya Khovanova, Jul 27 2007

			Differs from A002216 starting from a(6) = 36 = A002216(6) - 1: the polyhexes counted by a(6) do not include the ring-like configuration of 6 hexagons where one pair of hexagons which are adjacent from the planar point of view actually have an overlapping pair of external edges rather than a single shared edge. That non-planar configuration is shown in Fig. 2 of the Harary & Read (1970) reference in A002216.

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.

A. T. Balaban, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures, Tetrahedron, 44(1), 221-228 (1998). See Table 1.
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., 43 (2003), 842-851.
Gilles Caporossi and Pierre Hansen, Enumeration of Polyhex Hydrocarbons to h = 21, J. Chem. Inf. Comput. Sci., 38 (1998), 610-619.
Andrew Clarke, Isoperimetrical Polyhexes
Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron 42.19 (1986): 5291-5299. See Table 3.
J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Ratko Tošić, Dragan Mašulović, Ivan Stojmenović, Jon Brunvoll, Bjorg N. Cyvin and Sven J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., 35 (1995), 181-187.
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.
Eric Weisstein's World of Mathematics, Fusene.

Cf. A018190, A038143, A002216.
a(n) <= A000228(n), a(n) <= A057779(2n+1).
A131482 is the analog for polyominoes.

a(n) = A003104(n) + A323851(n). - Andrey Zabolotskiy, Feb 15 2023

a(11) from Tanya Khovanova, Jul 27 2007
a(12)-a(14) from John Mason, May 13 2021
a(15) from Trinajstić et al. (Table 4.2) added by Andrey Zabolotskiy, Feb 08 2023
a(16)-a(17) from Tošić et al., a(18)-a(20) from Caporossi & Hansen and a(21)-a(24) from Brinkmann, Caporossi & Hansen added by Andrey Zabolotskiy, Apr 11 2025

A108070 Number of fusenes with n hexagons.

Original entry on oeis.org

1, 1, 3, 7, 22, 82, 339, 1505, 7036, 33836, 166246, 829987, 4197273, 21456444, 110716585, 576027737, 3018986040, 15927330105, 84530870455, 451069339063, 2418927725532, 13030938290472, 70492771581350, 382816374644336, 2086362209298079, 11408580755666756

Offset: 1

Gunnar Brinkmann, Jun 05 2005

nonn

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.
Eric Weisstein's World of Mathematics, Fusene

Cf. A018190, A108071, A108072.

A038147 Number of polyhexes with n cells.

Original entry on oeis.org

1, 1, 3, 7, 22, 83, 341, 1519, 7114, 34350

Offset: 1

N. J. A. Sloane

J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 145.

See A000228 for another version of this sequence.

Cf. A018190, A038143-A038146.

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

A070765 Number of polyiamonds with n cells, without holes.

Original entry on oeis.org

1, 1, 1, 3, 4, 12, 24, 66, 159, 444, 1161, 3226, 8785, 24453, 67716, 189309, 528922, 1484738, 4172185, 11756354, 33174451, 93795220, 265565628, 753060469, 2138206966, 6078931114, 17302380313, 49302121747, 140627400927, 401510058179

Offset: 1

Joseph Myers, May 05 2002

If holes are allowed, we get A000577.

Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, Riddles of the sphinx tilings, arXiv:2304.14388 [cond-mat.stat-mech], 2023.
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.
Hai Pham-Van, Linh Tran-Phan-Thuy, Cuong Tran-Manh, Bich Do-Danh, and Hoang Luc-Huy, Two-Dimensional Clusters of Colloidal Particles Induced by Emulsion Droplet Evaporation, Nanomaterials (2020) Vol. 10, 156.

Equals A000577(n)-A070764(n). Cf. A000104, A018190.

More terms from Joseph Myers, Nov 11 2003

a(29) and a(30) from Joseph Myers, Nov 21 2010

cp's OEIS Frontend

A176638 Partial sums of A018190.

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

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A000104 Number of n-celled free polyominoes without holes.

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A258206 Number of strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotations and turning over.

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A070766 Number of polyhexes with n cells that tile the plane.

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A038142 Number of planar cata-polyhexes with n cells.

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A108070 Number of fusenes with n hexagons.

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A038147 Number of polyhexes with n cells.

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

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A070765 Number of polyiamonds with n cells, without holes.