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 1-3 of 3 results.

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.

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.

A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

0, 3, 12, 40, 98, 253, 540, 1199, 2415, 4893, 9268, 17864, 32421, 59265, 104632, 184338, 315414, 540155, 901845, 1504173, 2461932, 4013511, 6443170, 10314675, 16281749, 25608450, 39838855, 61716941, 94682665, 144726102
Offset: 1

Views

Author

Olivier Gérard, May 31 2024

Keywords

Comments

The length of each row is given by A000041.
As many sequences start like the nonnegative integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A001477 to have the same row sums for at least 8 terms: A089867, A089868, A089869, A089870, A118760, A123719, A130696, A136602, A254109, A258069, A258070, A258071, A266279, A272813, A273885, A273886, A273887, A273888.

Examples

			Illustration of the first few terms
.
0   | 0
3   | 1,  2
12  | 3,  4,  5
40  | 6,  7,  8,  9,  10
98  | 11, 12, 13, 14, 15, 16, 17
253 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28
540 | 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43
.

Crossrefs

Cf. A373300, original version, with positive integers A000027.
Cf. A001477, the nonnegative integers.
Cf. A027480, the sequence of row sums for a regular triangle.

Programs

  • Mathematica
    Module[{s = -1},
 Table[s +=
   PartitionsP[
    n - 1]; (s + PartitionsP[n]) (s + PartitionsP[n] - 1)/2 -
   s (s - 1)/2, {n, 1, 30}]]
Showing 1-3 of 3 results.

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