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.

0, 1, 3, 15, 3, 156, 15, 1284, 87, 642, 172, 2189, 149, 15, 2865, 215, 87

Offset: 0

Author

Pontus von Brömssen, Jan 24 2024

Keywords

• nonn
• frac
• more

Comments

For all n <= 16 except n = 7, 8, and 15, the optimal placement of the mine-free cells is unique up to rotations and reflections of the lattice (leaving the starting cell fixed). The three exceptional cases all have two optimal placements. For n = 7, the two optimal placements have the same underlying graph, but that is not the case for n = 8 and n = 15. 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 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.
		

Links

• Pontus von Brömssen, Illustration of the optimal mine-free cells for n = 1..16. (The random walk starts at the black dot.)
• Pontus von Brömssen, Plot of a(n)/A369371(n) vs n, using Plot2.

Crossrefs

Cf. A369371 (denominators), A366998 (square lattice), A369368 (hexagonal lattice).