A365964 -id:A365964 - cp's OEIS frontend

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

Pontus von Brömssen, Nov 01 2023

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.

			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.)

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.

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

A365963 Let I(n) be the moment of inertia of the polyomino with binary code A246521(n+1) about an axis through its center of mass perpendicular to the plane of the polyomino, the polyomino having a unit point mass in the center of each of its cells. a(n) is I(n) times the number of cells of the polyomino.

Original entry on oeis.org

0, 1, 4, 6, 14, 8, 11, 12, 20, 20, 32, 28, 38, 30, 32, 26, 26, 24, 30, 20, 50, 40, 49, 69, 61, 33, 49, 37, 46, 41, 52, 41, 53, 42, 61, 53, 52, 61, 34, 41, 50, 57, 85, 70, 73, 65, 69, 65, 60, 53, 56, 69, 49, 44, 45, 105, 82, 58, 64, 88, 86, 76, 74, 94, 86, 82

Offset: 1

Pontus von Brömssen, Sep 23 2023

If the cells have a uniform density of 1 instead of point masses in the centers, the moment of inertia is I(n) + k/6 = a(n)/k + k/6, where k is the number of cells.

			As an irregular triangle:
  0;
  1;
  4, 6;
  14, 8, 11, 12, 20;
  20, 32, 28, 38, 30, 32, 26, 26, 24, 30, 20, 50;
  ...
The five tetrominoes have moments of inertia 7/2, 2, 11/4, 3, 5 (in the order they appear in A246521). Multiplying these numbers by 4, we obtain the 4th row.
The last term of the k-th row of the irregular triangle corresponds to the straight k-omino, whose moment of inertia is k*(k^2-1)/12, so the last term of the k-th row is k^2*(k^2-1)/12 = A002415(k). (This ought to be the largest term of the k-th row.)

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).
Index entries for sequences related to moment of inertia.
Index entries for sequences related to polyominoes.

Cf. A000105 (row lengths), A002415, A246521, A365964 (row minima).

If the centers of the cells of the polyomino have coordinates (x_i,y_i), 1 <= i <= k, its moment of inertia is Sum_{i=1..k} x_i^2+y_i^2 - (Sum_{i=1..k} x_i)^2/k - (Sum_{i=1..k} y_i)^2/k.

cp's OEIS Frontend

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.

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