A007764 -id:A007764 - cp's OEIS frontend

A244088 Decimal expansion of 1/2+2/sqrt(13), a constant related to the asymptotic evaluation of the number of self-avoiding rook paths joining opposite corners on a 3 X n chessboard.

1, 0, 5, 4, 7, 0, 0, 1, 9, 6, 2, 2, 5, 2, 2, 9, 1, 2, 2, 0, 1, 8, 3, 4, 1, 7, 3, 3, 4, 5, 6, 9, 9, 9, 3, 7, 6, 3, 4, 6, 3, 5, 3, 3, 1, 9, 0, 5, 3, 1, 1, 4, 8, 0, 1, 9, 5, 5, 4, 5, 4, 3, 1, 6, 3, 4, 2, 6, 4, 1, 0, 6, 8, 9, 6, 8, 1, 5, 5, 4, 5, 3, 1, 0, 8, 4, 0, 2, 9, 3, 5, 6, 9, 5, 1, 5, 2, 4, 1, 8

Offset: 1

Jean-François Alcover, Jun 20 2014

			1.054700196225229122018341733456999376346353319...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.10.2 Rook paths on a chessboard, p. 334.

Cf. A006192, A007764, A007786, A007787, A244089.

RealDigits[1/2 + 2/Sqrt[13], 10, 100] // First

Asymptotic number of paths = p(k) ~ (1/2+2/sqrt(13)) * sqrt((3+sqrt(13))/2)^(2k), where k = n-1.

A244089 Decimal expansion of sqrt((3+sqrt(13))/2), a constant related to the asymptotic evaluation of the number of self-avoiding rook paths joining opposite corners on a 3 X n chessboard.

Original entry on oeis.org

1, 8, 1, 7, 3, 5, 4, 0, 2, 1, 0, 2, 3, 9, 7, 0, 6, 2, 0, 0, 7, 5, 1, 9, 4, 4, 8, 6, 0, 3, 5, 8, 2, 1, 9, 2, 6, 4, 6, 9, 4, 0, 3, 6, 4, 3, 1, 2, 7, 1, 3, 6, 1, 1, 2, 0, 6, 3, 3, 0, 7, 7, 0, 5, 8, 2, 7, 9, 8, 9, 9, 4, 3, 8, 6, 8, 3, 6, 5, 6, 9, 3, 6, 7, 8, 1, 9, 2, 0, 1, 7, 8, 1, 0, 0, 6, 2, 6, 7, 8

Offset: 1

Jean-François Alcover, Jun 20 2014

			1.8173540210239706200751944860358219264694...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.10.2 Rook paths on a chessboard, p. 334.

Cf. A006192, A007764, A007786, A007787, A244088.

RealDigits[Sqrt[(3 + Sqrt[13])/2], 10, 100] // First

Asymptotic number of paths = p(k) ~ (1/2+2/sqrt(13)) * sqrt((3+sqrt(13))/2)^(2k), where k = n-1.

A282425 The maximum number of steps Langton's ant can make confined to an n X n grid.

Original entry on oeis.org

0, 5, 16, 45, 84, 163, 260

Offset: 1

Rok Cestnik, Feb 14 2017

a(8) >= 338, a(9) >= 397, a(10) >= 502.
From Rok Cestnik, Aug 25 2017: (Start)
We are looking for the combination of grid configuration, ant orientation and ant position that yields the maximal number of steps before the ant leaves the grid. We consider all possible grid configurations and ant positions, but since the ant may move forward and backwards in time (see third considered symmetry below) we deduce that the maximal solution will always have the ant start from the edge of the grid.
For the sake of solution presentation, we consider these rules: at a white cell turn left, at a black cell turn right (vice versa results in the same behavior, just mirrored). Some cells might not get visited in a solution; therefore they are unconstrained, and we color then gray. We also take into consideration some symmetries of the ant to avoid presenting several maximal solutions that are just transformations of a single solution. That said, it is not impossible that two fundamentally different configurations would both have the same maximal number of steps.
Considered symmetries of the ant:
  1. rotational symmetry, e.g., we consider that the configuration
    +-----+-----+-----+              +-----+-----+-----+
    |  |  |BBBBB|     |              |     |BBBBB|     |
    |  v  |BBBBB|     |              |     |BBBBB| <-- |
    |     |BBBBB|     |              |     |BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |     |      is      |BBBBB|     |BBBBB|
    |BBBBB|     |     |  equivalent  |BBBBB|     |BBBBB|
    |BBBBB|     |     |      to      |BBBBB|     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |     |BBBBB|BBBBB|              |BBBBB|     |     |
    |     |BBBBB|BBBBB|              |BBBBB|     |     |
    |     |BBBBB|BBBBB|              |BBBBB|     |     |
    +-----+-----+-----+              +-----+-----+-----+
.
  2. mirror symmetry combined with color inversion, e.g., we consider that the configuration
    +-----+-----+-----+              +-----+-----+-----+
    |     |BBBBB|     |              |BBBBB|     |BBBBB|
    |     |BBBBB| <-- |              |B-->B|     |BBBBB|
    |     |BBBBB|     |              |BBBBB|     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |BBBBB|      is      |     |BBBBB|     |
    |BBBBB|     |BBBBB|  equivalent  |     |BBBBB|     |
    |BBBBB|     |BBBBB|      to      |     |BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |     |              |BBBBB|BBBBB|     |
    |BBBBB|     |     |              |BBBBB|BBBBB|     |
    |BBBBB|     |     |              |BBBBB|BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
.
  3. reversing the arrow of time combined with inverting the color of the cell on which the ant is located and turning the ant according to the color of its (now inverted) cell (with the chosen rules, if white turn left, if black turn right), e.g., the configuration
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    |B-->B|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |     |BBBBB|     |   will end   |BBBBB|BBBBB|BBBBB|
    |     |BBBBB|     |   in state   |BBBBB|BBBBB|BBBBB|
    |     |BBBBB|     |              |BBBBB|BBBBB|BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|BBBBB|     |              |     |     |BBBBB|
    |BBBBB|BBBBB|     |              | <-- |     |BBBBB|
    |BBBBB|BBBBB|     |              |     |     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
.
  and
.
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|BBBBB|BBBBB|              |     |     |BBBBB|
    |BBBBB|BBBBB|BBBBB|              |  ^  |     |BBBBB|
    |BBBBB|BBBBB|BBBBB|              |  |  |     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|BBBBB|BBBBB|   will end   |     |BBBBB|     |
    |BBBBB|BBBBB|BBBBB|   in state   |     |BBBBB|     |
    |BBBBB|BBBBB|BBBBB|              |     |BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|     |
    |BB^BB|     |BBBBB|              |BBBBB|BBBBB|     |
    |BB|BB|     |BBBBB|              |BBBBB|BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
.
  hence
.
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    |B-->B|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |     |BBBBB|     |      is      |BBBBB|BBBBB|BBBBB|
    |     |BBBBB|     |  equivalent  |BBBBB|BBBBB|BBBBB|
    |     |BBBBB|     |      to      |BBBBB|BBBBB|BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|BBBBB|     |              |BBBBB|     |BBBBB|
    |BBBBB|BBBBB|     |              |BB^BB|     |BBBBB|
    |BBBBB|BBBBB|     |              |BB|BB|     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
(End)
Due to the ant's complex nature, its trajectory is hard to predict; therefore, an exhaustive search through the possible grid configurations must be performed, making this sequence computationally demanding.

Rok Cestnik, Solution for n = 2
Rok Cestnik, Solution for n = 3
Rok Cestnik, Solution for n = 4
Rok Cestnik, Solution for n = 5
Rok Cestnik, Solution for n = 6
Rok Cestnik, Solution for n = 7
Rok Cestnik, C++ code for A282425
Rok Cestnik, Best found solution for n = 8
Rok Cestnik, Best found solution for n = 9
Rok Cestnik, Best found solution for n = 10
Wikipedia, Langton's ant

Cf. A007764, A099733, A255938.

a(7) from Rok Cestnik, Aug 12 2017

A356611 Number of SAWs spanning a rhomboidal domain of the hexagonal lattice.

Original entry on oeis.org

2, 50, 2256, 292006, 124394172, 182189852062, 937116505296162, 17167376550995687961, 1130911800993488803731078, 269650395624478266477331223678, 233772496350603982679550385266064014, 739330863241806743025423160490836132227125, 8551000409049037000098287028025432585191736309022

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D4.

Cf. A001006, A002026, A007764, A116485.

A356612 Number of SAPs crossing a rhomboidal domain of the hexagonal lattice.

Original entry on oeis.org

1, 3, 48, 3126, 775842, 727870836, 2575728525240, 34244061451559094, 1703999058661009145746, 316543880488539946466963896, 219157996022284922702859434801868, 564858713948847373563461482383973674774, 5415142061627863782256892670635702203299498106

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D5.

Cf. A001006, A002026, A007764, A116485.

A356613 Number of SAWs crossing a triangular domain of the hexagonal lattice.

Original entry on oeis.org

2, 7, 44, 515, 11500, 493704, 40751496, 6463642330, 1970190022696, 1154437344815284, 1300686960810345198, 2818300749120970598426, 11745284697899678209887246, 94153940687296424300453605522, 1451915619132744566900848537333082, 43072062058620235613855525243039798546

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..27
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D6.

Cf. A001006, A002026, A007764, A116485.

A356614 Number of SAWs crossing a triangular domain of the hexagonal lattice and including the top vertex.

Original entry on oeis.org

1, 3, 18, 210, 4716, 203130, 16781528, 2661898722, 811337884328, 475395297020430, 535618774376758222, 1160567857061063474508, 4836675324919658534327348, 38772333263059858336182467950, 597894854584620490267288203881970, 17736956492510173648327596231133813426

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D7.

Cf. A001006, A002026, A007764, A116485.

A356615 Number of SAPs crossing a triangular domain of the hexagonal lattice.

Original entry on oeis.org

1, 2, 9, 85, 1605, 59896, 4392639, 629739138, 175745776816, 95207239875508, 99934927799315359, 202993550188918062298, 797200289814680588454420, 6048794511036987586252009778, 88623124229469033988344357343229, 2506168305598107863294101582119745559

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D8.

Cf. A001006, A002026, A007764, A116485.

A053512 Half the number of non-self-intersecting paths from one corner of an n X n grid to the opposite corner.

Original entry on oeis.org

1, 6, 92, 4256, 631408, 287890282, 394680026626, 1633299243490821, 20522104351316248402, 784379015232375006607050, 91206645757124024620735442618, 32264019671635009481678592579241059, 34725332380760680832137350774453679498244

Offset: 2

Horst H. Manninger, Jan 14 2000

Cf. A007764.

a(n) = A007764(n) / 2. - Andrew Howroyd, May 12 2017

Corrected, additional terms from A007764 by Andrew Howroyd, May 12 2017

Offset corrected by Joerg Arndt, Dec 03 2018

A215577 Number of nonintersecting (or self-avoiding) rook paths joining opposite corner cells of an n X n X n grid, avoiding cells that are not on the surface.

Original entry on oeis.org

1, 18, 340812, 1553113040

Offset: 1

Alex Ratushnyak, Aug 16 2012

When n<3 there are n^3 cells available, otherwise n^3 - (n-2)^3.

The length of the step is 1. The length of the path varies.

Cf. A007764, A059783.

#include     // GCC -O3
char grid[4][4][4];
long long SIZE;
long long calc_ways(long long x, long long y, long long z) {
    long long n;
    if (grid[x][y][z]) return 0;
    if (x+y+z==SIZE*3-3) return 1;
    grid[x][y][z]=1;
    n=0;
    if (x>0)        n =calc_ways(x-1,y,z);  // go left
    if (x0)        n+=calc_ways(x,y-1,z);  // down
    if (y0)        n+=calc_ways(x,y,z-1);  // level down
    if (z

cp's OEIS Frontend

A244088 Decimal expansion of 1/2+2/sqrt(13), a constant related to the asymptotic evaluation of the number of self-avoiding rook paths joining opposite corners on a 3 X n chessboard.

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A244089 Decimal expansion of sqrt((3+sqrt(13))/2), a constant related to the asymptotic evaluation of the number of self-avoiding rook paths joining opposite corners on a 3 X n chessboard.

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A282425 The maximum number of steps Langton's ant can make confined to an n X n grid.

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A356611 Number of SAWs spanning a rhomboidal domain of the hexagonal lattice.

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A356612 Number of SAPs crossing a rhomboidal domain of the hexagonal lattice.

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A356613 Number of SAWs crossing a triangular domain of the hexagonal lattice.

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A356614 Number of SAWs crossing a triangular domain of the hexagonal lattice and including the top vertex.

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A356615 Number of SAPs crossing a triangular domain of the hexagonal lattice.

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A053512 Half the number of non-self-intersecting paths from one corner of an n X n grid to the opposite corner.

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A215577 Number of nonintersecting (or self-avoiding) rook paths joining opposite corner cells of an n X n X n grid, avoiding cells that are not on the surface.

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C