A003104 -id:A003104 - cp's OEIS frontend

A151516 Duplicate of A003104.

1, 1, 2, 4, 10, 24, 67, 182, 520, 1474, 4248, 12196

Offset: 1

dead

A038142 Number of planar cata-polyhexes with n cells.

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

A002013 Filaments with n square cells.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 13, 31, 65, 154, 347, 824, 1905, 4512, 10546, 24935, 58476, 138002, 323894, 763172, 1790585, 4213061, 9878541, 23214728, 54393063, 127687369, 298969219, 701171557, 1640683309, 3844724417, 8991137036, 21054243655, 49211076053

Offset: 0

N. J. A. Sloane

nonn

Or, number of 2-sided snake polyominoes with n cells. - Ed Pegg Jr, May 13 2009

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).
R. C. Tilley, R. G. Stanton and D. D. Cowan, The cell growth problem for filaments, pp. 310-339 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.

Arthur O'Dwyer, Polyomino strips, snakes, and ouroboroi, Dec 10 2022.
Ed Pegg, Jr., Illustrations of polyforms
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Polyomino

A333313 counts 2-sided (free) "strip" polyominoes; that is, snakes with no holes.

A003104 is the polyhex analog.

a(23) from Joseph Myers, Nov 22 2010
a(24)-a(26) from Sean A. Irvine, May 21 2013
a(27)-a(32) from John Mason, Dec 05 2021

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

A323851 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

Original entry on oeis.org

0, 0, 0, 1, 2, 12, 51, 229, 969, 4098, 16867, 68925, 278907, 1123302, 4507640

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.

A. T. Balaban, J. Brunvoll, B. N. Cyvin & S. J. Cyvin, Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures Tetrahedron, 44(1) (1988), 221-228. See Table 1.
Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron 42.19 (1986): 5291-5299. See Table 1, column C.

Cf. A003104, A323924-A323930.

a(13)-a(15) from Trinajstić et al. (Table 4.2, the number of cata-condensed benzenoids with h hexagons, branched) added by Andrey Zabolotskiy, Feb 08 2023

A323931 Unbranched catafusenes, mirror-symmetrical.

Original entry on oeis.org

0, 0, 1, 1, 4, 3, 12, 10, 34, 28, 97, 81, 271, 226, 764, 638, 2141, 1787, 6025, 5030, 16924, 14116, 47623, 39727, 133934, 111655, 377003, 314297, 1061064, 884199

Offset: 1

N. J. A. Sloane, Feb 09 2019

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) (1988), 221-228. See Table 1.
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.

The main sequences listed in Table 1 of Balaban et al. (1988) are A323931, A323932, A003104, A323851, A038142.

a(13)-a(30) from Cyvin, Brunvoll & Cyvin (Tables 14 & 15) added by Andrey Zabolotskiy, Feb 16 2023

A323924 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 22, 58

Offset: 1

N. J. A. Sloane, Feb 09 2019

Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron 42.19 (1986): 5291-5299. See Table 1.

Cf. A003104, A323923-A323930.

A323929 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

Original entry on oeis.org

0, 0, 1, 1, 7, 30, 141, 671, 3284, 15998, 78505, 385766, 2214783, 9382168

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.

Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron 42.19 (1986): 5291-5299. See Table 1.

Cf. A003104, A323923-A323930, A256981.

a(10)-a(14) from Trinajstić et al. (Table 4.11, the number of non-Kekuléan polyhexes with h hexagons) added by Andrey Zabolotskiy, Feb 08 2023

A383731 Number of hexagonal n-element polyominoes whose graph is a nonextensible path.

Original entry on oeis.org

2, 3, 17, 41, 140, 389, 1182, 3369, 9817, 27903, 79936, 226784, 645730, 1831574, 5204271, 14766828, 41938778, 119061270

Offset: 13

Bert Dobbelaere, May 07 2025

A 2-sided strip polyhex is a shape formed by a set of connected cells on a hexagonal grid, all cells having exactly two neighbors except for the first and last cell, which have only one neighbor. The shape is considered identical to its mirror image, as well as any translation or rotation of itself. This sequence counts the number of 2-sided strip polyhexes with n cells to which no additional cell can be added without introducing a branch or cycle. See also illustration.

Bert Dobbelaere, Illustration of initial terms colored by symmetry
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyhex

Cf. A003104.

A323925 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

Original entry on oeis.org

0, 0, 0, 0, 2, 12, 66, 331, 1678

Offset: 1

N. J. A. Sloane, Feb 09 2019

Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron 42.19 (1986): 5291-5299. See Table 1.

Cf. A003104, A323923-A323930.

cp's OEIS Frontend

A151516 Duplicate of A003104.

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A038142 Number of planar cata-polyhexes with n cells.

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A002013 Filaments with n square cells.

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

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A323851 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

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A323931 Unbranched catafusenes, mirror-symmetrical.

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A323924 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

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A323929 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

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A383731 Number of hexagonal n-element polyominoes whose graph is a nonextensible path.

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A323925 Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).

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