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Capped binary boundary codes for fusenes (all orientations and rotations included).
5

%I #27 Jun 11 2015 08:23:33

%S 1,127,1519,1783,1915,1981,2014,6007,7099,7645,7918,20335,22447,23479,

%T 23503,23995,24187,24253,24286,26551,27607,28123,28135,28381,28477,

%U 28510,29659,30187,30445,30451,30574,30622,31213,31477,31606,31609,31990,32122,32188

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

%C 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 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 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 Antti Karttunen, <a href="/A258012/b258012.txt">Table of n, a(n) for n = 0..20648</a>

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

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Helicene">Helicene</a>

%e 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 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 Cf. A254109, A258009.

%K nonn,base

%O 0,2

%A _Antti Karttunen_, May 31 2015