A343678 -id:A343678 - cp's OEIS frontend

A343746 The x,y,z coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

0, 0, 0, 0, 1, 2, 0, -1, 1, 0, 0, -1, 0, 1, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, -1, -1, 2, -1, -2, 1, -1, 0, -1, 0, 0, -1, 1, 2, 1, 2, 2, 2, 2, 0, 1, 0, 0, -1, -1, 0, -1, -2, 2, 1, -2, 3, 2, 0, 3, 2, -2, 2, 3, 0, 2, 3, -2, 1, 2, -2, 3, 2, -3, 1, 0, -4, 1, -2, -3, 1, -2, -2, 3, -3, 0, 3, -2, 2

Offset: 1

Scott R. Shannon, Apr 27 2021

See A343678 for the rules determining the knight's steps on the cubic lattice and for images of the resulting path.

			a(1),a(2),a(3) = 0,0,0. The knight starts at the origin on the lattice.
a(4),a(5),a(6) = 0,1,2. The 24 points the knight could step to on the first step all have only 1 visited neighbor and are all the same distance from the origin. Also they all have coordinates as arrangements of 0,+-1,+-2 thus the only way they can be separated is using rules 5 and 6 of A343678 which selects the smallest magnitudes of the x,y,z coordinates followed by the largest absolute x,y,z coordinates. This leads to the point (0,1,2) being selected.
a(7),a(8),a(9) = 0,-1,1. The other two possible points which have the same number of visited neighbors and are the same distance from the origin are (1,1,0) and (-1,1,0), but (0,-1,1) is chosen as that has the minimum x-coordinate magnitude.
a(52),a(53),a(54) = -1,-2,2. This is the first point that is chosen due to having the maximum product of the absolute values of its coordinate.
a(577),a(578),a(579) = -2,-3,-1. This is the first point that is stepped to that has two visited neighboring points.
a(1978),a(1979),a(1980) = -3,3,7. This is the first point that is stepped to that has three visited neighboring points.

Cf. A343678 (point square distances from origin), A343747 (point x coordinates), A343748 (point y coordinates), A343749 (point z coordinates) A330189, A329520, A316667.

A343747 The x coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -1, -1, 1, 2, 1, -1, -1, 1, 2, 2, 3, 3, 2, 2, 0, -2, -2, -3, -2, -2, 0, 0, 2, 2, 2, 2, 1, -1, -2, -2, -3, -3, -1, -1, -1, -3, -4, -4, -3, -1, -1, -1, -1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 0, -2, -2, -4, -4, -3, -3, -2, 0, 0, 0, 2, 3, 4, 4, 2, 0, -2, -2, -3, -3, -1, 0

Offset: 1

Scott R. Shannon, Apr 27 2021

See A343678 for the rules determining the knight's steps on the cubic lattice and for images of the resulting path.

Cf. A343678 (point square distances from origin), A343746 (point x,y,z coordinates), A343748 (point y coordinates), A343749 (point z coordinates), A330189, A329520, A316667.

A343748 The y coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

Original entry on oeis.org

0, 1, -1, 0, 1, -1, 1, 0, -1, -1, -1, 0, 1, 2, 2, 0, -1, -2, -2, 0, -2, 0, -2, -2, -3, -4, -3, -2, 0, 2, 3, 4, 3, 3, 2, 0, -2, -2, -2, 0, 2, 0, 2, 2, 3, 2, 1, 1, -1, -3, -3, -4, -3, -2, -3, -3, -1, 1, 2, 1, -1, -3, -1, -1, -1, -3, -3, -1, -1, 1, 1, 1, 3, 4, 3, 1, 1, 2, 3, 3, 3, 4, 4, 2, 3, 3, 3

Offset: 1

Scott R. Shannon, Apr 27 2021

See A343678 for the rules determining the knight's steps on the cubic lattice and for images of the resulting path.

A343678 (point square distances from origin), A343746 (point x,y,z coordinates), A343747 (point x coordinates), A343749 (point z coordinates)

A343749 The z coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

Original entry on oeis.org

0, 2, 1, -1, 1, 0, -1, 1, -1, -2, 0, 0, 2, 2, 0, 0, 0, 2, 3, 3, 2, 2, 1, 3, 1, 1, 1, 3, 3, 3, 1, 1, -1, 0, -2, -3, -2, -4, -3, -3, -2, -2, -1, -2, 0, 2, 2, 0, 1, 1, 2, 0, -2, -4, -4, -2, -3, -3, -1, 1, 2, 2, 3, 4, 3, 2, 1, 0, -2, -1, -3, -4, -3, -1, -1, -1, 1, 3, 3, 4, 3, 1, -1, -2, -2, -4, -3

Offset: 1

Scott R. Shannon, Apr 27 2021

See A343678 for the rules determining the knight's steps on the cubic lattice and for images of the resulting path.

A343678 (point square distances from origin), A343746 (point x,y,z coordinates), A343747 (point x coordinates), A343748 (point y coordinates).

A377015 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to a square which has been previously visited the fewest number of times. If two or more such squares exist the smallest numbered square is chosen.

Original entry on oeis.org

1, 10, 3, 6, 9, 4, 7, 2, 5, 8, 11, 14, 29, 32, 15, 12, 27, 24, 45, 20, 23, 44, 41, 18, 35, 38, 19, 16, 33, 30, 53, 26, 47, 22, 43, 70, 21, 40, 17, 34, 13, 28, 25, 46, 75, 42, 69, 104, 37, 62, 95, 58, 55, 86, 51, 48, 77, 114, 73, 108, 151, 68, 103, 64, 67, 36, 39, 66, 63, 96, 59, 56, 87, 52, 49, 78, 115, 74, 71, 106, 149, 102, 99, 140, 61, 94, 31, 54, 85, 50

Offset: 1

Scott R. Shannon, Nov 09 2024

Unlike similar sequences, e.g. A316667, A362027, A326922, in this variation the knight is never trapped as it can always move to the square which has been previously visited the fewest times, or if two or more surrounding squares exist with the same smallest previous visit count, then it can move to the smallest numbered square of these options.
The first 2016 terms are the same as A316667. In that sequence the path now ends, but here, as the knight is now surrounded by eight squares that have all been visited once, it now chooses the smallest numbered available square, 1733 in this case. This eventually leads it back toward the origin where it revisits the 1 starting square at a(2039). From here it once again chooses the surrounding square with the fewest previous visits, so it begins a new path, but it will avoid the path it previously took back to the origin since those squares will have two previous visits.
The above pattern repeats, causing the knight to go on various excursions of generally increasing length before it revisits the origin - see A377928 for the indices where a(n) = 1. The knight eventually tours paths of increasing complexity, in general moving along distorted loops which are defined by the previous visit count, with straight path lines between these loops. See the attached images.
Interestingly some of the paths between origin visits are identical; for example the path between the 2nd and 3rd visits to the origin is the same as that between the 32nd and 33rd visits. Likewise those between the 5th-6th and 23rd-24th, and 37th-38th and 40th-41st are the same.

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1, the default starting square.
a(2) = 10 as all eight surrounding available squares, 10, 12, 14, 16, 18, 20, 22, 24 have zero previous visits, so it chooses the smallest number of those, namely 10.
a(3) = 3 as there are seven available squares that have zero previous visits, and of those 3 is the smallest number. Note the 1 square is not considered as that has one previous visit which is more than the other seven squares.
a(2017) = 1733 as all eight surrounding available squares have been visit once, so it chooses the smallest number of those, namely 1733. This is the first term to differ from A316667.

Scott R. Shannon, Table of n, a(n) for n = 1..20000
Scott R. Shannon, Image of the first 100000 terms.
Scott R. Shannon, Image of the path between the first and second visit to the origin. In this and similar images, the steps are colored white, grey, red, orange, yellow, green, blue, indigo, and violet across the entire path to show their relative ordering. This image is the same as in A316667 except for the additional final steps, colored violet, back to the origin starting at a(2017). Also show as red circles are the visited squares whose eight available neighbours all have the same previous visit count.
Scott R. Shannon, Image of the path between the second and third visits to the origin.
Scott R. Shannon, Image of the path between the third and fourth visits to the origin. This shows the path beginning to get more complex - the knight travels along different loops of squares during different parts of its walk, moving between loops via generally straight line paths.
Scott R. Shannon, Image of the path between the sixth and seventh visits to the origin.
Scott R. Shannon, Image of the path between the eighth and ninth visits to the origin
Scott R. Shannon, Image of the path between the twelfth and thirteenth visits to the origin.
Scott R. Shannon, Image of the path between the twenty-third and twenty-fourth visits to the origin. This is identical to the path between the fifth and sixth visits.
Scott R. Shannon, Image of the path between the twenty-ninth and thirtieth visits to the origin.

Cf. A377928, A316667, A362027, A343678, A358278, A358150, A328929, A328909.

A377928 The indices k where A377015(k) = 1.

Original entry on oeis.org

1, 2039, 2703, 30083, 32155, 32437, 86925, 292101, 339137, 430611, 669371, 670563, 727051, 1161819, 1534325, 1541819, 1543011, 2027935, 2718001, 3266661, 3273829, 3730467, 4805861, 4806143, 5534871, 6371063, 7834735, 8926025, 9293575, 9664815, 12629449, 13645059, 13645723, 16510691, 19947389, 19952425, 22519739, 22520381, 24820941, 26657853, 26658495

Offset: 1

Scott R. Shannon, Nov 11 2024

nonn

These are the indices where a knight moving on a square spiral revisits the origin using the path rules given in A377015. See that sequence for further details.

Cf. A377015, A316667, A362027, A343678, A358278, A358150, A328929, A328909.

cp's OEIS Frontend

A343746 The x,y,z coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

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A343747 The x coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

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A343748 The y coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

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A343749 The z coordinates of the points visited by a knight on a 3D cubic lattice using the step rules given in A343678.

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A377015 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to a square which has been previously visited the fewest number of times. If two or more such squares exist the smallest numbered square is chosen.

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A377928 The indices k where A377015(k) = 1.

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