A317106 -id:A317106 - cp's OEIS frontend

A317108 Numbers missing from A317106.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128

Offset: 1

Daniël Karssen, Jul 24 2018

nonn

Equal to A299174 for n<=2930; a(2931)=5861, A299174(2931)=5862.
A317106 is finite, so this sequence is infinite.
See A317106 for further information.

Daniël Karssen, Table of n, a(n) for n = 1..67087

Cf. A299174, A317106.

A323469 On a spirally numbered square grid, with labels starting at 1, this is the number of steps that a (1,n) leaper makes before getting trapped, or -1 if it never gets trapped.

Original entry on oeis.org

-1, 2016, 3723, 13103, 14570, 26967, 101250, 158735, 132688, 220439, 144841, 646728, 350720, 66183, 75259, 248764, 118694, 307483, 238208, 189159, 139639, 183821, 151016, 171076, 114187, 262235, 178612, 257632, 124475, 178862, 143674, 196795, 60707, 309820

Offset: 1

N. J. A. Sloane, Jan 28 2019

sign

A (1,2) leaper is a chess knight.

a(2)-a(5) were computed by Daniël Karssen.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019)

The sequences involved in this set of related sequences are A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437, and A323469, A323470, A323471, A323472.

More terms from Rémy Sigrist, Jan 29 2019

A323471 On a spirally numbered square grid, with labels starting at 1, this is the number of the last cell that a (1,n) leaper reaches before getting trapped, or -1 if it never gets trapped.

Original entry on oeis.org

-1, 2084, 7081, 10847, 25963, 22421, 202891, 142679, 252953, 188501, 257479, 604328, 667827, 57217, 115497, 231930, 203331, 283651, 426851, 153521, 231299, 142267, 236487, 149872, 204527, 215033, 285983, 188082, 153461, 128802, 213853, 202259, 94967, 224778

Offset: 1

N. J. A. Sloane, Jan 28 2019

sign

A (1,2) leaper is a chess knight.

a(2)-a(5) were computed by Daniël Karssen.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019)

The sequences involved in this set of related sequences are A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437, and A323469, A323470, A323471, A323472.

More terms from Rémy Sigrist, Jan 29 2019

A317105 Squares visited by a (1,3)-leaper on a spirally numbered board and moving to the lowest available unvisited square at each step, squares labelled >=0.

Original entry on oeis.org

0, 26, 12, 4, 10, 6, 14, 8, 18, 2, 16, 32, 54, 84, 24, 28, 50, 46, 20, 38, 62, 94, 56, 86, 124, 48, 22, 40, 66, 34, 58, 88, 52, 30, 60, 36, 68, 42, 76, 112, 70, 44, 72, 106, 148, 100, 140, 188, 134, 180, 128, 90, 136, 96, 64, 104, 146, 98, 138, 92, 130, 176, 230

Offset: 0

Daniël Karssen, Jul 24 2018, following a suggestion from N. J. A. Sloane, Jul 15 2018

Board is numbered with the square spiral:
.
  16--15--14--13--12
   |               |
  17   4---3---2  11   .
   |   |       |   |
  18   5   0---1  10   .
   |   |           |
  19   6---7---8---9   .
   |
  20--21--22--23--24--25
.
The sequence is finite: at step 3722, square 7080 is visited, after which there are no unvisited squares within one move.

Daniël Karssen, Table of n, a(n) for n = 0..3722
Daniël Karssen, Figure showing the first 46 steps of the sequence
Daniël Karssen, Figure showing the complete sequence

Cf. A317106, A317107.

a(n) = A317106(n+1) - 1.

A323470 On a spirally numbered square grid, with labels starting at 0, this is the number of the final step that a (1,n) leaper makes before getting trapped, or -1 if it never gets trapped.

Original entry on oeis.org

-1, 2015, 3722, 13102, 14569, 26966, 101249, 158734, 132687, 220438, 144840, 646727, 350719, 66182, 75258, 248763, 118693, 307482, 238207, 189158, 139638, 183820, 151015, 171075, 114186, 262234, 178611, 257631, 124474, 178861, 143673, 196794, 60706, 309819

Offset: 1

N. J. A. Sloane, Jan 28 2019

sign

A (1,2) leaper is a chess knight.

a(2)-a(5) were computed by Daniël Karssen.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, C++ program for A323470
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019)

The sequences involved in this set of related sequences are A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437, and A323469, A323470, A323471, A323472.

More terms from Rémy Sigrist, Jan 29 2019

A323472 On a spirally numbered square grid, with labels starting at 0, this is the number of the last cell that a (1,n) leaper reaches before getting trapped, or -1 if it never gets trapped.

Original entry on oeis.org

-1, 2083, 7080, 10846, 25962, 22420, 202890, 142678, 252952, 188500, 257478, 604327, 667826, 57216, 115496, 231929, 203330, 283650, 426850, 153520, 231298, 142266, 236486, 149871, 204526, 215032, 285982, 188081, 153460, 128801, 213852, 202258, 94966, 224777

Offset: 1

N. J. A. Sloane, Jan 28 2019

sign

A (1,2) leaper is a chess knight.

a(2)-a(5) were computed by Daniël Karssen.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, Figure showing the complete figure for a (1, 624) leaper (where the color is function of the time)
Rémy Sigrist, C++ program for A323472
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019)

The sequences involved in this set of related sequences are A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437, and A323469, A323470, A323471, A323472.

More terms from Rémy Sigrist, Jan 29 2019

A306291 List of possible numbers for the final 'trapped' square of a knight moving on an infinitely large 2-dimensional spirally numbered board starting from any square.

Original entry on oeis.org

104, 125, 149, 150, 215, 235, 247, 260, 261, 262, 266, 277, 295, 311, 329, 330, 365, 368, 369, 385, 389, 404, 406, 408, 424, 425, 432, 445, 448, 467, 469, 489, 490, 494, 495, 512, 518, 534, 535, 536, 556, 557, 558, 561, 569, 580, 581, 582, 583, 586, 588, 604, 605, 606, 629, 631, 632, 634, 655, 659

Offset: 1

Scott R. Shannon, Feb 04 2019

This is a complete list of all the possible ending 'trapped' square values for the knight (2 by 1 leaper) starting from any square. The list has 1518 values - the knight starting from any square on the infinite board will eventually be trapped on a square with one of these numbers.
I do not have a proof this is the complete list of all ending values but I believe it is correct. I have checked every knight starting square up to 100000 and they all end on one of these 1518 squares. I then check further out to 110000 and ensure these paths always move inwards once they pass the square of values which contains the 100000 value, and check they do not move outwards again passed this square. As every knight sequence out to infinity would have to cross/land between this 100000 to 110000 group of values (as they are attracted toward square 1 due to their lowest-available-value preference), and as all values have been checked inside these, it implies all knight paths with starting square values out to infinity eventually end on this list of 1518 squares.
Also note this is the ordered sequence of all 1518 squares - the initial value found starting the knight at square 1 is 2084.

			The ending square for the knight starting on square with value 1 is 2084 (see A316667). The first starting square value to end on square 104 (the smallest value) is 175. The first starting square value to end on square 23134 (the largest value) is 11509.  Testing various upper limits has shown the square with number 404 is the most likely square for any random starting square to end on (about 8% of all sequences end on it). The complete list of 1518 end squares can be generated by checking all starting squares from 1 up to 17390 (which produces the 1518th different end square of value 16851).

Scott R. Shannon, Table of n, a(n) for n = 1..1518
Scott R. Shannon, Stripped down Java code for the sequence
Scott R. Shannon, Plot of all 1518 ending squares. The yellow line indicates the boundary of the 110000 starting values used in the generation of this data. In this spiral the square with value 2 is directly above the value 1 start square (green dot). The red dot is the square with value 404 (the most likely end value square).
Scott R. Shannon, Path for knight starting at square 175. The trapped square has value 104 - the smallest trapped square value. 8 Blue dots have been added around the final square to show all positions have been visited.
Scott R. Shannon, Path for knight starting at square 11509. The trapped square has value 23134 - the largest trapped square value. 8 Blue dots have been added around the final square to show all positions have been visited.
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019)

The sequences involved in this set of related sequences are A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437, and A323469, A323470, A323471, A323472.

A323749 Triangle read by rows: T(n,m) (1 <= n < m) = number of moves of a (m,n)-leaper (a generalization of a chess knight) until it can no longer move, starting on a board with squares spirally numbered from 1. Each move is to the lowest-numbered unvisited square. T(n,m) = -1 if the path never terminates.

Original entry on oeis.org

2016, 3723, 4634, 13103, 2016, 1888, 14570, 7574, 1323, 4286, 26967, 3723, 2016, 4634, 1796, 101250, 12217, 4683, 9386, 1811, 3487, 158735, 13103, 5974, 2016, 2758, 1888, 3984, 132688, 33864, 3723, 8900, 6513, 4634, 4505, 7796, 220439, 14570, 36232, 7574, 2016, 1323, 9052, 4286, 5679, 144841, 52738, 19370, 6355, 6425

Offset: 1

Jud McCranie, Jan 26 2019

The entries are the lower triangle of an array, for an (m,n)-leaper, where 1 <= n < m, ordered: (2,1), (3,1), (3,2), (4,1), (4,2), etc. Are all the paths finite? This appears to be an open question.

			A chess knight (a (2,1)-leaper) makes 2016 moves before it has no moves available (see A316667). Initial placement on square 1 counts as one move.

Jud McCranie, Table of n, a(n) for n = 1..19900
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Cf. A316667, A323750, A317106, A317471, A317416, A323750, A317438, A317916.

Edited by N. J. A. Sloane, Apr 30 2021

A323750 The label of the ending square of a (m,n)-leaper (a generalization of a chess knight) when it can no longer move, starting on a board with squares spirally numbered, starting at 1. Each move is to the lowest-numbered unvisited square.

Original entry on oeis.org

2084, 7081, 4698, 10847, 8399, 1164, 25963, 6760, 2269, 6500, 22421, 28273, 18946, 18643, 1202, 202891, 10059, 6425, 6662, 3039, 1383, 142679, 43325, 3744, 33725, 1460, 4639, 1952, 252953, 23684, 63577, 6040, 10841, 41836, 10017, 6338, 188501, 104413, 26546, 26967, 52736, 9145, 6580, 25799, 1869, 257479, 35652

Offset: 1

Jud McCranie, Jan 26 2019

The entries are the lower triangle of an array, for (m,n)-leaper, where 1 <= n < m, ordered: (2,1), (3,1), (3,2), (4,1), (4,2), etc.

			A chess knight (a (2,1)-leaper) makes 2016 moves before it reaches the square labeled 2084 and has no moves available (see A316667).

Jud McCranie, Table of n, a(n) for n = 1..19900
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Cf. A323749, A316667, A317106, A317471, A317416, A323750, A317438, A317916.

A306308 Table read by rows: the end square loops for a trapped knight moving on an infinitely large 2-dimensional spirally numbered board starting from any square.

Original entry on oeis.org

404, 3328, 2666, 1338, 736, 1535, 2168, 406, 2444, 2945, 2245, 605, 684, 2663, 2312, 3323, 935, 910

Offset: 1

Scott R. Shannon, Feb 05 2019

Construction: with a knight (a (1,2)-leaper) on an infinite spiral numbered board moving to the lowest numbered unvisited square (see A316884), start the knight on any square and continue moving it until it is trapped. Then start an entirely new sequence starting the knight at the same square at which it was previously trapped. Continue this process until the square at which the knight is trapped has occurred previously, indicating an end square loop. All starting squares for the knight on the infinite board will eventually lead to the knight path falling into one of the 3 end square loops listed here.

As the total number of squares in which the knight can be trapped is finite (see A306291), we expect there to be a finite number of end square loops - in theory, only those values (1518 is all) need to be checked when searching for an end square loop. However, all starting square values up to 302500 have been checked to determine into which of the 3 found loops the knight eventually falls. The 13-member loop with 406 as its lowest number is found to be the dominant loop, with about 89.6% of all initial starting squares going to it. The other 10.4% mostly go to the 3-member loop with 404 as its lowest number, with a decreasingly small remainder going to the 2-member loop with 910 as it lowest number. The attached 3-color image showing the start-value-to-loop mapping shows that the pattern of starting square to end square loops becomes very regular away from the center of the board.

			The 3 end square loops are:
1: 404, 3328, 2666
2: 1338, 736, 1535, 2168, 406, 2444, 2945, 2245, 605, 684, 2663, 2312, 3323
3: 935, 910
Starting the knight from the square 1 leads to the first 3-member loop after two iterations: the sequence of end squares is 2084, 404, 3328, 2666, 404, ... . Starting from the square 2 leads to the second (13-member) loop after ten iterations: the sequence is 711, 632, 4350, 3727, 3610, 7382, 2411, 4632, 4311, 1338, ... . The third (2-member) loop is not seen until the knight starts from square 284, the sequence being entered after two iterations via 1168, 935, 910, 935, ... .

Scott R. Shannon, Square positions for the 3 loops. The red line connects the 3 points of the first loop, the blue line connects the 13 points of the second loop, and the green line connects the 2 points of the third loop. The white point marks the central square with number 1.
Scott R. Shannon, Starting square to loop mapping. A plot of the first 302500 starting squares mapped via color to the end square loop into which the corresponding knight path eventually falls: red is the first (3-member) loop, blue the second (13-member) loop, green the third (2-member) loop. The white point marks the central square with number 1 for clarity (it actually falls into the red first loop).
Scott R. Shannon, The knight's path when starting at square 910. Showing path one of the 2-member loop - the green square is the starting square 910, the red square is the end square 935.
Scott R. Shannon, The knight's path when starting at square 935. Showing path two of the 2-member loop - the green square is the starting square 935, the red square is the end square 910.
Scott R. Shannon, Stripped down Java code to produce the loop values.
N. J. A. Sloane and Brady Haran, The Trapped Knight Numberphile video (2019).

Cf. A306291, A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437, and A323469, A323470, A323471, A323472.

cp's OEIS Frontend

A317108 Numbers missing from A317106.

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A323469 On a spirally numbered square grid, with labels starting at 1, this is the number of steps that a (1,n) leaper makes before getting trapped, or -1 if it never gets trapped.

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A323471 On a spirally numbered square grid, with labels starting at 1, this is the number of the last cell that a (1,n) leaper reaches before getting trapped, or -1 if it never gets trapped.

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A317105 Squares visited by a (1,3)-leaper on a spirally numbered board and moving to the lowest available unvisited square at each step, squares labelled >=0.

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A323470 On a spirally numbered square grid, with labels starting at 0, this is the number of the final step that a (1,n) leaper makes before getting trapped, or -1 if it never gets trapped.

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A323472 On a spirally numbered square grid, with labels starting at 0, this is the number of the last cell that a (1,n) leaper reaches before getting trapped, or -1 if it never gets trapped.

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A306291 List of possible numbers for the final 'trapped' square of a knight moving on an infinitely large 2-dimensional spirally numbered board starting from any square.

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A323750 The label of the ending square of a (m,n)-leaper (a generalization of a chess knight) when it can no longer move, starting on a board with squares spirally numbered, starting at 1. Each move is to the lowest-numbered unvisited square.

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A306308 Table read by rows: the end square loops for a trapped knight moving on an infinitely large 2-dimensional spirally numbered board starting from any square.

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