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.

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

Offset: 1

Author

Hugo Pfoertner, Oct 03 2024

Keywords

• nonn
• frac

Comments

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.

Examples

			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, ...
		

Links

• 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.

Crossrefs

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.

Programs

PARI

\\ 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))