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A258012 Capped binary boundary codes for fusenes (all orientations and rotations included).

%I A258012 #28 Feb 16 2025 08:33:25
%S A258012 1,127,1519,1783,1915,1981,2014,6007,7099,7645,7918,20335,22447,23479,
%T A258012 23503,23995,24187,24253,24286,26551,27607,28123,28135,28381,28477,
%U A258012 28510,29659,30187,30445,30451,30574,30622,31213,31477,31606,31609,31990,32122,32188
%N A258012 Capped binary boundary codes for fusenes (all orientations and rotations included).
%C A258012 Differs from A258002 for the first time at n=6622, where a(6622) = 69131119 which is missing from A258002 because that number codes for one of the 26 different orientations of the same 26-edge six-hex polyhex where the two hexes at the ends of the pattern touch each other. This pattern is isomorphic to benzenoid [6]Helicene (up to chirality, see the illustrations at Wikipedia-page).
%C A258012 The terms in this sequence are those whose binary representation can be rewritten to 127 (in binary "1111111", which encodes the boundary of a single hexagon) with an appropriate sequence of invocations of recurrences A254109 and A258009. However, there are some intricacies as how this should be done to get correct results. (Please see Kovič paper.)
%C A258012 Note that the papers in literature employ different, "Boundary Edges Code for Benzenoid Systems" (BEC for short) but to which these binary boundary codes can be directly related via their run-lengths.
%H A258012 Antti Karttunen, <a href="/A258012/b258012.txt">Table of n, a(n) for n = 0..20648</a>
%H A258012 Guo, Hansen, Zheng, <a href="http://dx.doi.org/10.1016/S0166-218X(01)00180-9">Boundary uniqueness of fusenes</a>, Discrete Applied Mathematics 118 (2002), pp. 209-222.
%H A258012 A. Karttunen, <a href="http://web.archive.org/web/20071021150215/http://ndirty.cute.fi/~karttu/matikka/Prolog/polyhexp.txt">Related ideas coded in Prolog around 2004 - 2006</a> (at Internet Archive. Might contain a few erroneous definitions.)
%H A258012 Jurij Kovič, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match72/n1/match72n1_27-38.pdf">How to Obtain The Number of Hexagons in a Benzenoid System from Its Boundary Edges Code</a>, MATCH Commun. Math. Comput. Chem. 72 (2014) pp. 27-38.
%H A258012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fusene.html">Fusene</a>
%H A258012 Wikipedia, <a href="http://en.wikipedia.org/wiki/Helicene">Helicene</a>
%e A258012 8167737748888 is included in the sequence, as it encodes a 42-edge polyhex pattern which is composed of two seven-hex "crowns" connected by a snake-like "S-piece".
%Y A258012 Subsequences: A258002 (only strictly non-overlapping codes, i.e., the holeless polyhexes), A258013 (only the lexicographically largest representatives from each equivalence class obtained by rotating).
%Y A258012 Cf. A254109, A258009.
%K A258012 nonn,base
%O A258012 0,2
%A A258012 _Antti Karttunen_, May 31 2015