A376609 a(n) is the numerator of the expected number of random moves of a chess king to reach a position outside an nXn chessboard, starting in one of the corners.

1, 8, 72, 46, 23747, 94968, 12161644, 158536576, 165181795263, 1779861954248, 60921563004721184, 136512657826472304, 38548316743830620183051, 581371653539561314, 2630585854108441990301102856, 120104329127347395409698056, 5092493809189909792181005355935991197, 6666722670813237580783418910187983288

Offset: 1

Author

Hugo Pfoertner, Oct 03 2024

Keywords

• nonn
• frac
• nice

Comments

The king visits the Moore neighborhood (see A272763). 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, 8/5, 72/35, 46/19, 23747/8723, 94968/31879, 12161644/3797647, 158536576/46627015, 165181795263/46174521031, ...
Approximately 1, 1.6, 2.057, 2.421, 2.722, 2.979, 3.202, 3.400, 3.577, 3.738, ...
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..45
• Hugo Pfoertner, Plot of A376609(n)/A376610(n) vs n, using Plot 2.
• Hugo Pfoertner, Results of a simulation of 10^9 walks on the 8X8 board.

Crossrefs

A376610 are the corresponding denominators.

A376606 and A376607 are similar for a rook walk with unit steps.

A376736 and A376737 are similar for a chess knight.

Cf. A140518, A272763, A323140, A323561, A323562.

Programs

PARI

\\ Uses function droprob from A376606
kingmoves = [[1, 0], [0, 1], [0, -1], [-1, 0], [-1, -1], [-1, 1], [1, -1], [1, 1]];
a376609(n) = numerator(droprob(n,kingmoves,8))