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Number of planar cata-polyhexes with n cells.
10

%I #35 Feb 16 2023 14:55:18

%S 1,1,2,5,12,36,118,411,1489,5572,21115,81121,314075,1224528,4799205

%N Number of planar cata-polyhexes with n cells.

%C Number of cata-condensed benzenoid hydrocarbons with n hexagons.

%C a(n) is the number of n-celled polyhexes with perimeter 4n+2. 4n+2 is the maximal perimeter of an n-celled polyhex. a(n) is the number of n-celled 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 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 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 Andrew Clarke, <a href="http://www.recmath.com/PolyPages/PolyPages/index.htm?IsopolyH.htm">Isoperimetrical Polyhexes</a>

%H 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polyhex.html">Polyhex</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fusene.html">Fusene</a>.

%F a(n) = A003104(n) + A323851(n). - _Andrey Zabolotskiy_, Feb 15 2023

%Y Cf. A018190, A038143.

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

%K nonn,hard,more

%O 1,3

%A _N. J. A. Sloane_

%E a(11) from _Tanya Khovanova_, Jul 27 2007

%E a(12)-a(14) from _John Mason_, May 13 2021

%E a(15) from Trinajstić et al. (Table 4.2) added by _Andrey Zabolotskiy_, Feb 08 2023