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.
0, 1, 4, 12, 28, 8, 124, 128, 263, 9, 1303, 519707, 380, 3435
Offset: 0
Examples
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.)
Links
- Pontus von Brömssen, Illustration of the optimal mine-free points for n = 1..13. (The random walk starts at the black dot.)
- Pontus von Brömssen, Plot of a(n)/A366999(n) vs n, using Plot2.
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