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A258012 Capped binary boundary codes for fusenes (all orientations and rotations included). 5
1, 127, 1519, 1783, 1915, 1981, 2014, 6007, 7099, 7645, 7918, 20335, 22447, 23479, 23503, 23995, 24187, 24253, 24286, 26551, 27607, 28123, 28135, 28381, 28477, 28510, 29659, 30187, 30445, 30451, 30574, 30622, 31213, 31477, 31606, 31609, 31990, 32122, 32188 (list; graph; refs; listen; history; text; internal format)



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).

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.)

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.


Antti Karttunen, Table of n, a(n) for n = 0..20648

Guo, Hansen, Zheng, Boundary uniqueness of fusenes, Discrete Applied Mathematics 118 (2002), pp. 209-222.

A. Karttunen, Related ideas coded in Prolog around 2004 - 2006 (at Internet Archive. Might contain a few erroneous definitions.)

Jurij Kovič, How to Obtain The Number of Hexagons in a Benzenoid System from Its Boundary Edges Code, MATCH Commun. Math. Comput. Chem. 72 (2014) pp. 27-38.

Eric Weisstein's World of Mathematics, Fusene

Wikipedia, Helicene


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".


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).

Cf. A254109, A258009.

Sequence in context: A300339 A189026 A287653 * A258002 A025598 A115153

Adjacent sequences:  A258009 A258010 A258011 * A258013 A258014 A258015




Antti Karttunen, May 31 2015



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Last modified May 27 16:48 EDT 2022. Contains 354110 sequences. (Running on oeis4.)