cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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
Offset: 0

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Author

Antti Karttunen, May 31 2015

Keywords

Comments

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.

Examples

			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".

Links

Crossrefs

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.

A254109 If n <= 63, a(n) = n; for n > 63: a(32n + 14) = 8*n + 5, a(64n + 30) = 4*n + 3, and for other cases with n > 63: a(2n) = 2*a(n), a(2n+1) = 2*a(n) + 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 21, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 7
Offset: 0

Views

Author

Antti Karttunen, Mar 11 2015

Keywords

Comments

This sequence is a rewriting-recurrence which attempts to contract the perimeter of binary boundary coded holeless polyhexes and other fusenes by 2 or 4 edges, where first possible (from the least significant end of n), and if no such contraction is possible, then it fixes n. Together with recurrence A258009 can be used to obtain the terms of A258012, please see comments there.

Examples

			The first term where a(n) is different from n occurs at n=78, as 78 = "1001110" in binary, where the clause a(32n + 14) = 8*n + 5 will rewrite the trailing "01110" part as "101", resulting binary string "10101" = 21 in decimal.

Links

Crossrefs

Cf. A255561, A255568, A258009, A258012.

Formula

If n <= 63, a(n) = n; for n > 63: a(32n + 14) = 8*n + 5, a(64n + 30) = 4*n + 3, and for other cases with n > 63: a(2n) = 2*a(n), a(2n+1) = 2*a(n) + 1.

Extensions

Recurrence corrected to match the intended usage by Antti Karttunen, Jun 05 2015
Showing 1-2 of 2 results.

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