A212715 -id:A212715 - cp's OEIS frontend

A323559 Number of rooted self-avoiding knight's paths of length n on an infinite chessboard with first move specified.

1, 7, 49, 337, 2323, 15805, 107737, 727619, 4921655, 33056939, 222323989, 1487064391, 9957971965, 66391431607, 443085643919, 2946553003837, 19611967535129, 130149475953673

Offset: 1

Hugo Pfoertner, Jan 17 2019

Cf. A001411, A212715, A323131, A272773, A323140, A323560.

A376736 a(n) is the numerator of the expected number of random moves of a chess knight to reach a position outside an nXn chessboard, starting in one of the corners.

Original entry on oeis.org

1, 1, 4, 62, 269, 1766, 395497, 101338, 44125237, 227721959, 3361699348115, 483866477194862, 277887411827604127, 790848403160840410, 2785714552717079970073201, 89715505143567836216964174, 2034961072108249587083318018747, 457177774768288408431166142758841, 1085703228381446052419019696184520372520

Offset: 1

Hugo Pfoertner, Oct 03 2024

The piece does not pay attention to its position and will fall off the board if it makes a move beyond the edge of the board.

			1, 1, 4/3, 62/43, 269/167, 1766/1017, 395497/213488, 101338/51901, 44125237/21578387, 227721959/106983448, ...
Approximately 1, 1, 1.333, 1.442, 1.611, 1.736, 1.853, 1.953, 2.045, 2.129, 2.206, ...

Hugo Pfoertner, Table of n, a(n) for n = 1..45
Hugo Pfoertner, Plot of A376736(n)/A376737(n) vs n, using Plot 2.
Hugo Pfoertner, Results of a simulation of 10^9 walks on the 8X8 board.

A376737 are the corresponding denominators.
A376606 and A376607 are similar for a rook walk with unit steps.
A376609 and A376610 are similar for a chess king.
Cf. A003192, A212715, A323131, A323132, A323559, A323560.

\\ Uses function droprob from A376606
knightmoves = [[2, 1], [1, 2], [-1, 2], [-2, 1], [-2, -1], [-1, -2], [1, -2], [2, -1]];
a376736(n) = numerator(droprob(n, knightmoves, 8))

A323561 Number of rooted self-avoiding king's walks of n moves on an infinite chessboard with first move specified.

Original entry on oeis.org

2, 14, 92, 584, 3644, 22482, 137626, 837466, 5072590, 30611376, 184171252, 1105262004, 6618842522, 39564403462, 236123357538, 1407249202976, 8376673823516

Offset: 1

Hugo Pfoertner, Jan 17 2019

The first move is either (0,0) -> (1,0) or (0,0) -> (1,1). Rotated paths are not counted separately.

Cf. A001411, A212715, A272773, A323131, A323140, A323559, A323562.

A289204 Number of (undirected) paths in the n X n knight graph.

Original entry on oeis.org

0, 0, 56, 14980, 19005336, 278982789260

Offset: 1

Eric W. Weisstein, Jun 28 2017

Eric Weisstein's World of Mathematics, Graph Path
Eric Weisstein's World of Mathematics, Knight Graph

Cf. A234623, A212715, A165134, A001230.

a(5)-a(6) from Andrew Howroyd, Jul 01 2017

A347363 Number of self-avoiding knight's paths from the lower left corner to the lower right corner of a 3 X n chessboard.

Original entry on oeis.org

1, 0, 2, 8, 32, 156, 871, 5292, 28702, 154162, 845532, 4662014, 25579463, 140098348, 767973001, 4212065280, 23097682805, 126643657272, 694390484065, 3807499106946, 20877386149018, 114474503105178, 627683328355315, 3441701959286326, 18871492466212538

Offset: 1

Andrzej Kukla, Aug 29 2021

If we enumerate the squares in the 3 X n board like this:
------------------------------------
| 1 | 4 | 7 | 10 | 13 | ... | 3n-2 |
------------------------------------
| 2 | 5 | 8 | 11 | 14 | ... | 3n-1 |
------------------------------------
| 3 | 6 | 9 | 12 | 15 | ... |  3n  |
------------------------------------
then a(n) is the number of self-avoiding knight's paths on such a board from square 3 to square 3n.

			For n = 4 we have exactly 8 self-avoiding paths starting at square 3 and ending at square 12:
  3,  4,  9, 10,  5, 12;
  3,  4,  9,  2,  7, 12;
  3,  8,  1,  6,  7, 12;
  3,  4, 11,  6,  7, 12;
  3,  8,  1,  6, 11,  4,  9,  2,  7, 12;
  3,  4, 11,  6,  7,  2,  9, 10,  5, 12;
  3,  8,  1,  6,  7,  2,  9, 10,  5, 12;
  3,  8,  1,  6, 11,  4,  9, 10,  5, 12;

Cf. A118067, A169696, A212715.

a(8)-a(15) from Pontus von Brömssen, Aug 30 2021

Terms a(16) and beyond from Andrew Howroyd, Nov 19 2021

cp's OEIS Frontend

A323559 Number of rooted self-avoiding knight's paths of length n on an infinite chessboard with first move specified.

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A376736 a(n) is the numerator of the expected number of random moves of a chess knight to reach a position outside an nXn chessboard, starting in one of the corners.

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A323561 Number of rooted self-avoiding king's walks of n moves on an infinite chessboard with first move specified.

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A289204 Number of (undirected) paths in the n X n knight graph.

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A347363 Number of self-avoiding knight's paths from the lower left corner to the lower right corner of a 3 X n chessboard.

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