A368387 - OEIS

%I #11 Jan 12 2024 15:02:33

%S 1,1,3,3,35,35,35,35,35,154,462,462,231,462,231,462,924,462,462,7,924,

%T 1846572,492573081,19019,19019,5073,19019,1804297,7379372,492573081,

%U 7379372,1804297,19019,1846572,19019,5534529,7379372,19019,492573081,5534529,7379372,19019,19019,7379372,19019,5534529,19019,19019,14758744,5534529,7379372,19019,5534529,44276232,1844843,19019

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

%C 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. A368386(n)/a(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).

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

%H Pontus von Brömssen, <a href="/A368387/b368387.txt">Table of n, a(n) for n = 1..6473</a> (rows 1..10).

%H Persi Diaconis and William Fulton, <a href="http://www.seminariomatematico.polito.it/rendiconti/cartaceo/49-1/95.pdf">A growth model, a game, an algebra, Lagrange inversion, and characteristic classes</a>, Rend. Semin. Mat. Univ. Politec. Torino, Vol. 49 (1991), No. 1, 95-119.

%H Gregory F. Lawler, Maury Bramson, and David Griffeath, <a href="https://doi.org/10.1214/aop/1176989542">Internal diffusion limited aggregation</a>, The Annals of Probability 20 no. 4 (1992), 2117-2140.

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.

%F A368386(n)/a(n) = (A368392(n)/A368393(n))*A335573(n+1).

%e As an irregular triangle:

%e     1;

%e     1;

%e     3,   3;

%e    35,  35,  35,  35,  35;

%e   154, 462, 462, 231, 462, 231, 462, 924, 462, 462, 7, 924;

%e   ...

%e There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.

%e For three squares, the probability for an L (or right) tromino (whose binary code is 7 = A246521(4)) is 2/3, so a(3) = 3. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 3.

%Y Cf. A000105, A246521, A335573, A367672, A367761, A367995, A368386 (numerators), A368389, A368391, A368392, A368393, A368660 (external diffusion-limited aggregation).

%K nonn,frac,tabf

%O 1,3

%A _Pontus von Brömssen_, Dec 22 2023