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.

Showing 1-4 of 4 results.

A258013 Capped binary boundary codes for fusenes, only the maximal representatives of each equivalence class obtained by rotating.

Original entry on oeis.org

1, 127, 2014, 7918, 31606, 32122, 32188, 126394, 127930, 128476, 486838, 503254, 503482, 505306, 505564, 506332, 511450, 511462, 511708, 511804, 513514, 513772, 513778, 514540, 514804, 514936, 2012890, 2012902, 2013916, 2021098, 2021212, 2022124, 2025196, 2039254, 2043610, 2043622, 2045674, 2045788, 2046700
Offset: 0

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Author

Antti Karttunen, May 31 2015

Keywords

Comments

A258017(n) gives the count of terms with binary width 2n + 1.
Differs from A258003 for the first time at n=875, which here contains a(875) = 131821024 the smallest polyhex (26 edges, six hexes) where two hexes (at the opposite ends of a coiled pattern) meet to touch each other.
This pattern is isomorphic to benzenoid [6]Helicene (up to chirality, see the illustrations at Wikipedia-page).
Note that here, in contrast to "Boundary Edges Code for Benzenoid Systems" (see links at A258012), if a fusene has no bilateral symmetry then both variants of the corresponding one-sided fusene (their codes) are included in this sequence, the other obtained from the other by turning it over.

Links

Crossrefs

Subsequences: A258003, A258015.
Intersection of A257250 and A258012.
Cf. A258014 (same codes without the most significant bit).
Cf. also A258017.

A258002 Capped binary boundary codes for holeless strictly non-overlapping polyhexes (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, 80815, 81271, 89527, 89551, 89719, 93655, 93883, 95191, 95707, 95719, 95965, 96061
Offset: 0

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Author

Antti Karttunen, May 16 2015

Keywords

Comments

The sequence consists of those terms of A255571 whose every A080541/A080542-rotation is also a term of A255571 and in their binary representation the number of 1's is larger than the number of 0's. More precisely, after the initial term a(0)=1 (which stands for an empty path) each term has seven more 1's than 0's in their binary representation, i.e., A037861(a(n)) = -7 for all n >= 1.

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

Intersection of A072600 and A258001.
Intersection of A255571 and A258012.
Subsequence: A258003 (lexicographically largest representatives).
Cf. A037861.
Differs from A258012 for the first time at n=6622.

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

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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

A258009 If n <= 16, a(n) = n; for n > 16: a(16n + 6) = 16*n + 9, and for other cases with n > 16: 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, 25, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 41, 39, 40, 41, 42, 43, 50, 51, 46, 47, 48, 49, 50, 51, 52, 53, 57, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 73, 71, 72, 73, 74, 75, 82, 83, 78, 79, 80, 81, 82, 83, 84, 85, 89, 87, 100
Offset: 0

Views

Author

Antti Karttunen, May 31 2015

Keywords

Comments

Function a(n) rewrites in the binary representation of n the rightmost occurrence of substring "0110" to "1001", provided at least one such substring is present, otherwise fixes n.
The values 224694 and 486838 shown in the example section are capless and one-capped binary boundary codes for seven-hex polyhex-configuration called "crown" (the name employed for example in Guo et al paper) and the resulting values are the respective codes for one hex smaller polyhexes. The crown is one of the polyhexes that are too round that the related recurrence A254109 could make any dent in their boundary. Together with the latter can be used to obtain the terms of A258012, please see comments there.

Examples

			For n = 224694 ("110110110110110110" in binary) we rewrite the rightmost "0110" to "1001" resulting "110110110110111001" in binary, which is 224697 in decimal, thus a(224694) = 224697.
For n = 486838 ("1110110110110110110" in binary) we rewrite the rightmost "0110" to "1001" resulting "1110110110110111001" in binary, which is 486841 in decimal, thus a(486838) = 486841.

Links

Crossrefs

Cf. A254109, A258012.

Formula

If n <= 16, a(n) = n; for n > 16: a(16n + 6) = 16*n + 9, and for other cases with n > 16: a(2n) = 2*a(n), a(2n+1) = 2*a(n) + 1.
Showing 1-4 of 4 results.

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