A368660 - OEIS

%I #17 Jan 24 2024 16:43:53

%S 1,0,1,0,0,0,0,0,5,0,0,0,7,4,0,0,0,2,2,4,0,0,0,6,7,2,0,0,0,0,8,3,6,5,

%T 2,0,0,0,7,1,4,4,0,1,0,0,0,4,2,9,6,4,5,1,0,0,0,8,5,3,2,3,1,6,1,0,0,0,

%U 9,1,9,9,0,7,2,3,0,0,0,0,0,0,5,4,0,7,7,2,6,0,0

%N Square array read by antidiagonals; the n-th row is the decimal expansion of the probability that the free polyomino with binary code A246521(n+1) appears in diffusion-limited aggregation on the square lattice.

%C Given the current set of cells in a diffusion-limited aggregation process on the square lattice, with new cells coming in from infinity, the probability that the next cell appears in a given position can be found by "Spitzer's recipe" (see Spitzer (1976) and Wolf (1991)). These probabilities can then be aggregated to probabilities for each polyomino to appear.

%C Each row corresponds to a number in the field Q(Pi), i.e., a number of the form (Sum_{i=0..j} p_i*Pi^i)/(Sum_{i=0..k} q_i*Pi^i), with p_i and q_i integers.

%C Rows A130866(k-1)+1 to A130866(k) correspond to k-celled polyominoes, k >= 2. The sum of the numbers on those rows is 1.

%D Frank Spitzer, Principles of Random Walk, 2nd edition, Springer, 1976. See Chapter III.

%H Pontus von Brömssen, <a href="/A368660/b368660.txt">Table of n, a(n) for n = 1..1596</a> (first 56 antidiagonals).

%H Pontus von Brömssen, <a href="/A368660/a368660.txt">First 12 decimal digits and exact values of the form (Sum_{i=0..j} p_i*Pi^i)/(Sum_{i=0..k} q_i*Pi^i) for rows 1..56</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Diffusion-limited_aggregation">Diffusion-limited aggregation</a>.

%H Marek Wolf, <a href="https://doi.org/10.1103/PhysRevA.43.5504">Hitting probabilities of diffusion-limited-aggregation clusters</a>, Physical Review A 43 (1991), 5504-5517; <a href="https://www.researchgate.net/publication/13383995_Hitting_probabilities_of_diffusion-limited-aggregation_clusters">ResearchGate link</a>.

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

%e Array begins:

%e   1.00000000000000000000... (monomino)

%e   1.00000000000000000000... (domino)

%e   0.57268748908837848701... (L tromino)

%e   0.42731251091162151298... (I tromino)

%e   0.42649395750130487018... (L tetromino)

%e   0.05462942885357382723... (square tetromino)

%e   0.20430093094721062115... (T tetromino)

%e   0.15177943827373482673... (S tetromino)

%e   0.16279624442417585468... (I tetromino)

%e   0.13219133154126607406... (P pentomino)

%e   0.06837364801045779482... (V pentomino)

%e   0.03733461160442202363... (W pentomino)

%e   0.14605587435506817264... (L pentomino)

%e   0.15786504558818518196... (Y pentomino)

%e   0.10529476741119453953... (N pentomino)

%e   0.04279427184030725060... (U pentomino)

%e   0.08270007323598911231... (T pentomino)

%e   0.10865945602909460112... (F pentomino)

%e   0.04929714951722524019... (Z pentomino)

%e   0.01279646275569121440... (X pentomino)

%e   0.05663730811109879467... (I pentomino)

%e   ...

%Y Cf. A000105, A130866, A246521, A368661, A368662, A368863 (fixed polyominoes).

%Y Rows 3-9 are A368663, A368664, A368665, A368666, A368667, A368668, A368669.

%Y Corresponding sequences for internal diffusion-limited aggregation: A368386, A368387.

%K nonn,tabl,cons

%O 1,9

%A _Pontus von Brömssen_, Jan 02 2024