A323132 -id:A323132 - cp's OEIS frontend

A323131 Number of uncrossed rooted knight's paths of length n on an infinite board.

1, 7, 47, 303, 1921, 11963, 74130, 454484, 2779152, 16882278, 102384151, 618136584, 3727827148, 22408576099, 134595908277, 806452390868

Offset: 1

Hugo Pfoertner, Jan 05 2019

The direction of the first move is kept fixed.

The average number of steps of a random walk using such knight moves with forbidden crossing is 45 (compare to A322831).

			a(1) = 1: The fixed initial move.
a(2) = 7: Relative to the direction given by the initial move, there are 7 possible direction changes. The backward direction is illegal for the self-avoiding uncrossed path. Only for the right angle turn its mirror image would coincide with the turn in the opposite direction. Therefore this move would be eliminated in the unrooted walks, making a(2) > A323132(2) = 6.
a(3) = 47: 2 of all 7*7 = 49 continuation moves already lead to a crossing of the first path segment.
See illustrations at Pfoertner link.

Hugo Pfoertner, Illustrations of rooted uncrossed knight's paths of length <= 3, (2019).

Cf. A003192, A272773, A323132, A323133, A323134, A323559.

Erroneous (as pointed out by Bert Dobbelaere) a(8) and a(10) corrected by Hugo Pfoertner, Jan 18 2019

a(12)-a(16) from Bert Dobbelaere, Jan 18 2019

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))

A323133 Number of symmetric uncrossed unrooted knight's paths of length n on an infinite board.

Original entry on oeis.org

1, 6, 7, 29, 46, 170, 299, 969

Offset: 1

Hugo Pfoertner, Jan 05 2019

A path is considered as symmetric if its "spine", i.e., the connection of the end points of the moves by straight lines, has mirror or point symmetry. The non-symmetric details of a single move are ignored.

			See Pfoertner link.

Hugo Pfoertner, Illustrations of symmetric uncrossed knight's paths of length <= 6, (2019).

Cf. A003192, A306176, A306178, A323131, A323132, A323134.

cp's OEIS Frontend

A323131 Number of uncrossed rooted knight's paths of length n on an infinite board.

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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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A323133 Number of symmetric uncrossed unrooted knight's paths of length n on an infinite board.

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