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

1,2

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.

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.

Example

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, ...

Prog

(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))

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.

Sequence in context: A162155 A163344 A064743 * A109868 A153521 A153650

Adjacent sequences: A376603 A376604 A376605 * A376607 A376608 A376609

Keyword

nonn,frac

Author

Ruediger Jehn and Hugo Pfoertner, Oct 03 2024

Status

approved