cp's OEIS Frontend

A369368 Numerator of the maximum expected number of steps of a random walk on the cells of the hexagonal lattice before it lands on a mined cell, given that all but n cells are mined.

%I A369368 #10 Jan 28 2024 09:20:52
%S A369368 0,1,6,3,24,165,2550,10,3090,390,1296,265230
%N A369368 Numerator of the maximum expected number of steps of a random walk on the cells of the hexagonal lattice before it lands on a mined cell, given that all but n cells are mined.
%C A369368 For all n <= 11, the optimal placement of the mine-free cells is unique up to rotations and reflections of the lattice (leaving the starting cell fixed).
%H A369368 Pontus von Brömssen, <a href="/A369368/a369368.svg">Illustration of the optimal mine-free cells for n = 1..11</a>. (The random walk starts at the black dot.)
%H A369368 Pontus von Brömssen, <a href="https://oeis.org/plot2a?name1=A369368&amp;name2=A369369&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Plot of a(n)/A369369(n) vs n</a>, using Plot2.
%e A369368 For n = 0, the random walk stops before it can take any step, so a(0) = 0.
%e A369368 For n = 1, only the starting cell can be swept, so the random walk always stops after 1 step and a(1) = 1.
%e A369368 For n = 2, we can sweep the starting cell and one adjacent cell. The random walk then has probability 1/6 of surviving at each step, which implies that the expected number of steps is 6/5, so a(2) = 6. (The number of steps follows a geometric distribution.)
%e A369368 For n = 3, the best strategy is to sweep three mutually adjacent cells. As for n = 2, the number of steps follows a geometric distribution, now with the probability 1/3 of surviving at each step, so the expected number of steps is 3/2 and a(3) = 3.
%e A369368 See linked illustration for optimal solutions for 1 <= n <= 11.
%Y A369368 Cf. A369369 (denominators), A366998 (square lattice), A369370 (triangular lattice).
%K A369368 nonn,frac,more
%O A369368 0,3
%A A369368 _Pontus von Brömssen_, Jan 24 2024