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.

0, 1, 6, 3, 24, 165, 2550, 10, 3090, 390, 1296, 265230

Offset: 0

Author

Pontus von Brömssen, Jan 24 2024

Keywords

• nonn
• frac
• more

Comments

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).

Examples

			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 1/6 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.)
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.
See linked illustration for optimal solutions for 1 <= n <= 11.
		

Links

• Pontus von Brömssen, Illustration of the optimal mine-free cells for n = 1..11. (The random walk starts at the black dot.)
• Pontus von Brömssen, Plot of a(n)/A369369(n) vs n, using Plot2.

Crossrefs

Cf. A369369 (denominators), A366998 (square lattice), A369370 (triangular lattice).