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

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more

Comments

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

Examples

			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.
		

References

• 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.

Links

• 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.

Crossrefs

Cf. A018190, A038143, A002216.

a(n) <= A000228(n), a(n) <= A057779(2n+1).

A131482 is the analog for polyominoes.

Formula

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

Extensions

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