cp's OEIS Frontend

A038142 Number of planar cata-polyhexes with n cells.

%I A038142 #47 Apr 14 2025 11:45:39
%S A038142 1,1,2,5,12,36,118,411,1489,5572,21115,81121,314075,1224528,4799205,
%T A038142 18896981,74695032,296275836,1178741568,4702507923,18806505243,
%U A038142 75380203150,302754225098,1218239791106
%N A038142 Number of planar cata-polyhexes with n cells.
%C A038142 Number of cata-condensed benzenoid hydrocarbons with n hexagons.
%C A038142 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
%D A038142 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.
%H A038142 A. T. Balaban, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, <a href="https://doi.org/10.1016/S0040-4020(01)85110-3">Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures</a>, Tetrahedron, 44(1), 221-228 (1998). See Table 1.
%H A038142 Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, <a href="https://doi.org/10.1021/ci025526c">A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons</a>, J. Chem. Inf. Comput. Sci., 43 (2003), 842-851.
%H A038142 Gilles Caporossi and Pierre Hansen, <a href="https://doi.org/10.1021/ci970116n">Enumeration of Polyhex Hydrocarbons to h = 21</a>, J. Chem. Inf. Comput. Sci., 38 (1998), 610-619.
%H A038142 Andrew Clarke, <a href="http://www.recmath.com/PolyPages/PolyPages/index.htm?IsopolyH.htm">Isoperimetrical Polyhexes</a>
%H A038142 Wenchen He and Wenjie He, <a href="https://doi.org/10.1016/S0040-4020(01)82078-0">Generation and enumeration of planar polycyclic aromatic hydrocarbons</a>, Tetrahedron 42.19 (1986): 5291-5299. See Table 3.
%H A038142 J. V. Knop et al., <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match16/match16_119-134.pdf">On the total number of polyhexes</a>, Match, No. 16 (1984), 119-134.
%H A038142 Ratko Tošić, Dragan Mašulović, Ivan Stojmenović, Jon Brunvoll, Bjorg N. Cyvin and Sven J. Cyvin, <a href="https://doi.org/10.1021/ci00024a002">Enumeration of polyhex hydrocarbons to h = 17</a>, J. Chem. Inf. Comput. Sci., 35 (1995), 181-187.
%H A038142 N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, <a href="https://doi.org/10.1351/pac198855020379">Computer Generation of Isomeric Structures</a>, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
%H A038142 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyhex.html">Polyhex</a>.
%H A038142 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fusene.html">Fusene</a>.
%F A038142 a(n) = A003104(n) + A323851(n). - _Andrey Zabolotskiy_, Feb 15 2023
%e A038142 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.
%Y A038142 Cf. A018190, A038143, A002216.
%Y A038142 a(n) <= A000228(n), a(n) <= A057779(2n+1).
%Y A038142 A131482 is the analog for polyominoes.
%K A038142 nonn,hard,more
%O A038142 1,3
%A A038142 _N. J. A. Sloane_
%E A038142 a(11) from _Tanya Khovanova_, Jul 27 2007
%E A038142 a(12)-a(14) from _John Mason_, May 13 2021
%E A038142 a(15) from Trinajstić et al. (Table 4.2) added by _Andrey Zabolotskiy_, Feb 08 2023
%E A038142 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