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A366998 a(n) is the numerator of the maximum expected number of steps of a random walk on the square lattice until it lands on a mined lattice point, given that mines are placed on all but n points.

%I A366998 #19 Jan 28 2024 09:20:48
%S A366998 0,1,4,12,28,8,124,128,263,9,1303,519707,380,3435
%N A366998 a(n) is the numerator of the maximum expected number of steps of a random walk on the square lattice until it lands on a mined lattice point, given that mines are placed on all but n points.
%C A366998 For all n <= 13 except n = 3, the optimal placement of the mine-free points is unique, up to rotations and reflections with respect to the starting point. See linked illustration.
%H A366998 Pontus von Brömssen, <a href="/A366998/a366998.svg">Illustration of the optimal mine-free points for n = 1..13</a>. (The random walk starts at the black dot.)
%H A366998 Pontus von Brömssen, <a href="https://oeis.org/plot2a?name1=A366998&amp;name2=A366999&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Plot of a(n)/A366999(n) vs n</a>, using Plot2.
%e A366998 For n = 0, the random walk stops before it can take any step, so a(0) = 0.
%e A366998 For n = 1, only the mine at the starting point can be swept, so the random walk always stops after 1 step and a(1) = 1.
%e A366998 For n = 2, the starting point and one adjacent point can be swept. The random walk then has probability 1/4 of surviving at each step, which implies that the expected number of steps is 4/3, so a(2) = 4. (The number of steps follows a geometric distribution.)
%Y A366998 Cf. A365964, A366999 (denominators), A369368 (hexagonal lattice), A369370 (triangular lattice).
%K A366998 nonn,frac,more
%O A366998 0,3
%A A366998 _Pontus von Brömssen_, Nov 01 2023