A376606 a(n) is the numerator of the expected number of moves to reach a position outside an nXn chessboard, starting in one of the corners, when performing a random walk with unit steps on the square lattice.

1, 2, 11, 10, 99, 122, 619, 4374, 187389, 482698, 11031203, 33386106, 32723853563, 139832066, 150236161755, 633573154269934, 5755694771977, 189378719187729770, 509943025510535499, 6031948951257694364778, 1044408374351599765540157091, 27891966006517951087819226666

Offset: 1

Author

Ruediger Jehn and Hugo Pfoertner, Oct 03 2024

Keywords

• nonn
• frac

Comments

The moves are that of chess rook with moves of unit length or of a chess king restricted to the Von Neumann 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, 2, 11/4, 10/3, 99/26, 122/29, 619/136, 4374/901, 187389/36562, 482698/89893, ...
Approximately 1, 2, 2.75, 3.333, 3.808, 4.207, 4.551, 4.855, 5.125, 5.370, 5.593, ...
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..45
• Hugo Pfoertner, Plot of A376606(n)/A376607(n) vs n, using Plot 2.

Crossrefs

A376607 are the corresponding denominators.

A376609 and A376610 are similar for a chess king visiting the Moore neighborhood.

A376736 and A376737 are similar for a chess knight.

Cf. A001263, A007764, A272763.

Programs

PARI

droprob(n,moves=[[1,0],[0,1],[0,-1],[-1,0]], nmoves=4) = {my(np=n^2+1, M=matrix(np), P=1/nmoves); for(t=1, nmoves, for( i=1, n, my(ti=i+moves[t][1]); for(j=1,n,my(tj=j+moves[t][2]); my(m=(i-1)*n+j); if(ti<1 || ti>n || tj<1 || tj>n, M[m,np]+=P, my(mt=(ti-1)*n+tj); M[m,mt]+=P)))); vecsum((1/(matid(np)-M))[,1])};
a376606(n) = numerator(droprob(n))