A038142 -id:A038142 - cp's OEIS frontend

A323933 Erroneous version of A038142.

1, 1, 2, 5, 12, 36, 116, 411, 1489, 5572, 21115, 81121

Offset: 1

dead

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

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

A131482 a(n) is the number of n-celled polyominoes with perimeter 2n+2.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 83, 255, 847, 2829, 9734, 33724, 118245, 416816, 1478602, 5267171, 18840144, 67611472, 243378415, 878407170, 3178068821, 11523323634, 41865833602, 152382134767

Offset: 1

Tanya Khovanova, Jul 27 2007

2n+2 is the maximal perimeter of an n-celled polyomino. a(n) is the number of n-celled polyominoes 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).

Andrew Clarke, Isoperimetrical Polyominoes, The Poly Pages.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).

Cf. A000105, A057730. Diagonal of A342243.
A359522 counts only polyominoes with holes.
A002013 counts only unbranched polyominoes.
A038142 is the analog for polyhexes.

a(n) <= A000105(n), a(n) <= A057730(n+1).

a(n) >= A000602(n) [see comment on edge graph trees]. - R. J. Mathar, Mar 08 2021

a(14)-a(16) from David Radcliffe, Dec 25 2017
a(17) from David Radcliffe, Dec 26 2017
a(18)-a(24) from John Mason, Dec 11 2021

A038143 Hexagonal cata-condensed helicenes with n cells (non-planar cata-fused benzenoid hydrocarbons).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 35, 200, 1121, 5919, 30509, 153187, 756825, 3688195

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.

Equals A002216-A038142. Cf. A018190.

a(12)-a(15) from Andrey Zabolotskiy, Feb 16 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

A066331 Number of fixed hexagonal polyominoes with n cells and tree-like structure.

Original entry on oeis.org

1, 3, 9, 29, 99, 348, 1260, 4644, 17382, 65822, 251655, 969819, 3762517, 14680890, 57567228, 226712655, 896252850, 3555116583, 14144563158

Offset: 1

Jan Kristian Haugland, Jan 01 2002

Sean A. Irvine, Java program (github)

Cf. A001207 (all polyhexes), A038142 (free tree-like).

a(16)-a(19) from Sean A. Irvine, Oct 08 2023

A131488 a(n) is the number of polyhexes with n edges, including inner edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 1, 5, 0, 1, 3, 6, 12, 3, 4, 14, 26, 39, 10, 25, 70, 116, 139, 67, 152, 347, 514, 567, 414, 884, 1744, 2408, 2561, 2498, 4967

Offset: 1

Tanya Khovanova, Jul 28 2007

An n-celled polyhex with perimeter p has (6n+p)/2 edges. The maximum number of edges in an n-celled polyhex is 5n+1.

Given Clarke's table T(p,n), a(n) is an antidiagonal sum selecting entries in a (1,3)-leaper's moves. - R. J. Mathar, Feb 23 2021

			a(31) = T(p=26,A=6) + T(p=20,A=7) = 36+3 = 39. a(34) = T(p=26,A=7) + T(p=20,A=8) = 69+1 = 70. a(35) = 107+9. a(36) = 118+21. a(41) = 411+155+1. a(44) = 1621 +123. a(45) = 1986+420+2. a(46) = 1489+1046+26. - _R. J. Mathar_, Feb 23 2021

Andrew Clarke, Isoperimetrical Polyhexes

Cf. A000228: Number of hexagonal polyominoes (or planar polyhexes) with n cells. A057779: Hexagonal polyominoes (or polyhexes, A000228) with perimeter 2n. A038142: Number of planar cata-polyhexes with n cells. A131487: analog for square tiling.

Extended to a(48). - R. J. Mathar, Feb 23 2021

A130623 Number of polyhexes with perimeter at most 2n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 8, 20, 34, 84, 182

Offset: 1

Tanya Khovanova, Aug 11 2007

The perimeter of a polyhex is always even.
Partial sums of A057779.
a(2n+1) >= A131467(n).

Cf. A038142 (planar cata-polyhexes with n cells).
Cf. A057779 (hexagonal polyominoes with perimeter 2n).
Cf. A131467 (planar polyhexes with at most n cells).

Offset corrected by John Mason, Jan 16 2023

A137193 Number of n-celled polyhexes with perimeter < 4n+2.

Original entry on oeis.org

0, 0, 1, 2, 10, 46, 215, 1037, 5083, 24918, 122437

Offset: 1

Tanya Khovanova, Mar 03 2008

These are n-cell polyhexes with the perimeter less than the maximum possible one.

If we associate a graph to a polyhex with vertices representing cells and edges representing two cells with a common edge, then this sequence enumerates polyhexes whose corresponding graphs have cycles.

A000228(n)-A038142(n)

A176638 Partial sums of A018190.

Original entry on oeis.org

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).

cp's OEIS Frontend

A323933 Erroneous version of A038142.

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

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A131482 a(n) is the number of n-celled polyominoes with perimeter 2n+2.

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A038143 Hexagonal cata-condensed helicenes with n cells (non-planar cata-fused benzenoid hydrocarbons).

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A323931 Unbranched catafusenes, mirror-symmetrical.

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A066331 Number of fixed hexagonal polyominoes with n cells and tree-like structure.

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A131488 a(n) is the number of polyhexes with n edges, including inner edges.

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A130623 Number of polyhexes with perimeter at most 2n.

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A137193 Number of n-celled polyhexes with perimeter < 4n+2.

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A176638 Partial sums of A018190.

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