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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.

%I A376609 #18 Oct 05 2024 07:51:18
%S A376609 1,8,72,46,23747,94968,12161644,158536576,165181795263,1779861954248,
%T A376609 60921563004721184,136512657826472304,38548316743830620183051,
%U A376609 581371653539561314,2630585854108441990301102856,120104329127347395409698056,5092493809189909792181005355935991197,6666722670813237580783418910187983288
%N 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.
%C A376609 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.
%H A376609 Hugo Pfoertner, <a href="/A376609/b376609.txt">Table of n, a(n) for n = 1..45</a>
%H A376609 Hugo Pfoertner, <a href="/plot2a?name1=A376609&amp;name2=A376610&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Plot of A376609(n)/A376610(n) vs n</a>, using Plot 2.
%H A376609 Hugo Pfoertner, <a href="/A376609/a376609_1.png">Results of a simulation of 10^9 walks on the 8X8 board</a>.
%e A376609 1, 8/5, 72/35, 46/19, 23747/8723, 94968/31879, 12161644/3797647, 158536576/46627015, 165181795263/46174521031, ...
%e A376609 Approximately 1, 1.6, 2.057, 2.421, 2.722, 2.979, 3.202, 3.400, 3.577, 3.738, ...
%o A376609 (PARI) \\ Uses function droprob from A376606
%o A376609 kingmoves = [[1, 0], [0, 1], [0, -1], [-1, 0], [-1, -1], [-1, 1], [1, -1], [1, 1]];
%o A376609 a376609(n) = numerator(droprob(n,kingmoves,8))
%Y A376609 A376610 are the corresponding denominators.
%Y A376609 A376606 and A376607 are similar for a rook walk with unit steps.
%Y A376609 A376736 and A376737 are similar for a chess knight.
%Y A376609 Cf. A140518, A272763, A323140, A323561, A323562.
%K A376609 nonn,frac,nice
%O A376609 1,2
%A A376609 _Hugo Pfoertner_, Oct 03 2024