A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

1, 1, 2, 1, 8, 4, 17, 4, 2, 57, 5, 5, 5, 73, 5, 5, 73, 73, 5, 1, 5, 49321, 28165117, 20, 20, 338, 20, 246038, 63425, 28165117, 63425, 123019, 20, 49321, 20, 149998, 63425, 20, 117209258, 74999, 63425, 10, 20, 63425, 20, 74999, 10, 10, 63425, 149998, 63425, 10, 149998, 5000341, 64770, 5

Offset

1,3

Comments

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. a(n)/A368387(n) is the probability that, when the appropriate number of cells have been added, those cells form the free polyomino with binary code A246521(n+1).

Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).

Persi Diaconis and William Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Semin. Mat. Univ. Politec. Torino, Vol. 49 (1991), No. 1, 95-119.

Gregory F. Lawler, Maury Bramson, and David Griffeath, Internal diffusion limited aggregation, The Annals of Probability 20 no. 4 (1992), 2117-2140.

Index entries for sequences related to polyominoes.

Formula

a(n)/A368387(n) = (A368392(n)/A368393(n))*A335573(n+1).

Example

As an irregular triangle:
   1;
   1;
   2, 1;
   8, 4, 17, 4,  2;
  57, 5,  5, 5, 73, 5, 5, 73, 73, 5, 1, 5;
  ...
There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
For three squares, the probability for an L (or right) tromino (whose binary code is 7 = A246521(4)) is 2/3, so a(3) = 2. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 1.

Crossrefs

Cf. A000105, A246521, A335573, A367671, A367760, A367994, A368387 (denominators), A368388, A368390, A368392, A368393, A368660 (external diffusion-limited aggregation).

Sequence in context: A223550 A178102 A245836 * A135520 A136230 A193892

Adjacent sequences: A368383 A368384 A368385 * A368387 A368388 A368389

Keyword

nonn,frac,tabf

Author

Pontus von Brömssen, Dec 22 2023

Status

approved