cp's OEIS Frontend

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.

%I A368386 #12 Jan 12 2024 15:02:16
%S A368386 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,
%T A368386 338,20,246038,63425,28165117,63425,123019,20,49321,20,149998,63425,
%U A368386 20,117209258,74999,63425,10,20,63425,20,74999,10,10,63425,149998,63425,10,149998,5000341,64770,5
%N 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.
%C A368386 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).
%C A368386 Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.
%H A368386 Pontus von Brömssen, <a href="/A368386/b368386.txt">Table of n, a(n) for n = 1..6473</a> (rows 1..10).
%H A368386 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 A368386 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 A368386 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%F A368386 a(n)/A368387(n) = (A368392(n)/A368393(n))*A335573(n+1).
%e A368386 As an irregular triangle:
%e A368386    1;
%e A368386    1;
%e A368386    2, 1;
%e A368386    8, 4, 17, 4,  2;
%e A368386   57, 5,  5, 5, 73, 5, 5, 73, 73, 5, 1, 5;
%e A368386   ...
%e A368386 There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
%e A368386 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.
%Y A368386 Cf. A000105, A246521, A335573, A367671, A367760, A367994, A368387 (denominators), A368388, A368390, A368392, A368393, A368660 (external diffusion-limited aggregation).
%K A368386 nonn,frac,tabf
%O A368386 1,3
%A A368386 _Pontus von Brömssen_, Dec 22 2023