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

Author

Pontus von Brömssen, Nov 01 2023

Keywords

• nonn
• frac
• more

Comments

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.

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.

Crossrefs

Cf. A365964, A366999 (denominators), A369368 (hexagonal lattice), A369370 (triangular lattice).