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A376736 a(n) is the numerator of the expected number of random moves of a chess knight to reach a position outside an nXn chessboard, starting in one of the corners.

%I A376736 #14 Oct 05 2024 07:46:33
%S A376736 1,1,4,62,269,1766,395497,101338,44125237,227721959,3361699348115,
%T A376736 483866477194862,277887411827604127,790848403160840410,
%U A376736 2785714552717079970073201,89715505143567836216964174,2034961072108249587083318018747,457177774768288408431166142758841,1085703228381446052419019696184520372520
%N A376736 a(n) is the numerator of the expected number of random moves of a chess knight to reach a position outside an nXn chessboard, starting in one of the corners.
%C A376736 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 A376736 Hugo Pfoertner, <a href="/A376736/b376736.txt">Table of n, a(n) for n = 1..45</a>
%H A376736 Hugo Pfoertner, <a href="/plot2a?name1=A376736&amp;name2=A376737&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Plot of A376736(n)/A376737(n) vs n</a>, using Plot 2.
%H A376736 Hugo Pfoertner, <a href="/A376736/a376736_1.png">Results of a simulation of 10^9 walks on the 8X8 board</a>.
%e A376736 1, 1, 4/3, 62/43, 269/167, 1766/1017, 395497/213488, 101338/51901, 44125237/21578387, 227721959/106983448, ...
%e A376736 Approximately 1, 1, 1.333, 1.442, 1.611, 1.736, 1.853, 1.953, 2.045, 2.129, 2.206, ...
%o A376736 (PARI) \\ Uses function droprob from A376606
%o A376736 knightmoves = [[2, 1], [1, 2], [-1, 2], [-2, 1], [-2, -1], [-1, -2], [1, -2], [2, -1]];
%o A376736 a376736(n) = numerator(droprob(n, knightmoves, 8))
%Y A376736 A376737 are the corresponding denominators.
%Y A376736 A376606 and A376607 are similar for a rook walk with unit steps.
%Y A376736 A376609 and A376610 are similar for a chess king.
%Y A376736 Cf. A003192, A212715, A323131, A323132, A323559, A323560.
%K A376736 nonn,frac
%O A376736 1,3
%A A376736 _Hugo Pfoertner_, Oct 03 2024