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.
Original entry on oeis.org
0, 1, 4, 12, 28, 8, 124, 128, 263, 9, 1303, 519707, 380, 3435
Offset: 0
For n = 0, the random walk stops before it can take any step, so a(0) = 0.
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.
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.)
A369370
Numerator of the maximum expected number of steps of a random walk on the cells of the triangular lattice before it lands on a mined cell, given that all but n cells are mined.
Original entry on oeis.org
0, 1, 3, 15, 3, 156, 15, 1284, 87, 642, 172, 2189, 149, 15, 2865, 215, 87
Offset: 0
For n = 0, the random walk stops before it can take any step, so a(0) = 0.
For n = 1, only the starting cell can be swept, so the random walk always stops after 1 step and a(1) = 1.
For n = 2, we can sweep the starting cell and one adjacent cell. The random walk then has probability 1/3 of surviving at each step, which implies that the expected number of steps is 3/2, so a(2) = 3. (The number of steps follows a geometric distribution.)
For n = 3, the best strategy is to sweep the starting cell and two of its neighboring cells. Let x be the expected length of the random walk with the given starting cell, and let y be the expected length of a random walk starting at one of the other two cells. By conditioning on the first step, it follows that the equations x = 1 + y*2/3 and y = 1 + x/3 hold, giving x = 15/7 and a(3) = 15.
See linked illustration for optimal solutions for 1 <= n <= 16.
A369369
Denominator 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.
Original entry on oeis.org
1, 1, 5, 2, 13, 74, 961, 3, 871, 103, 319, 61289
Offset: 0
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