login
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.
7
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, 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.
Sequence in context: A162155 A163344 A064743 * A109868 A153521 A153650
KEYWORD
nonn,frac
AUTHOR
Ruediger Jehn and Hugo Pfoertner, Oct 03 2024
STATUS
approved