A368863 -id:A368863 - cp's OEIS frontend

A368660 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.

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

Offset: 1

Pontus von Brömssen, Jan 02 2024

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.
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.
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.

			Array begins:
  1.00000000000000000000... (monomino)
  1.00000000000000000000... (domino)
  0.57268748908837848701... (L tromino)
  0.42731251091162151298... (I tromino)
  0.42649395750130487018... (L tetromino)
  0.05462942885357382723... (square tetromino)
  0.20430093094721062115... (T tetromino)
  0.15177943827373482673... (S tetromino)
  0.16279624442417585468... (I tetromino)
  0.13219133154126607406... (P pentomino)
  0.06837364801045779482... (V pentomino)
  0.03733461160442202363... (W pentomino)
  0.14605587435506817264... (L pentomino)
  0.15786504558818518196... (Y pentomino)
  0.10529476741119453953... (N pentomino)
  0.04279427184030725060... (U pentomino)
  0.08270007323598911231... (T pentomino)
  0.10865945602909460112... (F pentomino)
  0.04929714951722524019... (Z pentomino)
  0.01279646275569121440... (X pentomino)
  0.05663730811109879467... (I pentomino)
  ...

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

Pontus von Brömssen, Table of n, a(n) for n = 1..1596 (first 56 antidiagonals).
Pontus von Brömssen, 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.
Wikipedia, Diffusion-limited aggregation.
Marek Wolf, Hitting probabilities of diffusion-limited-aggregation clusters, Physical Review A 43 (1991), 5504-5517; ResearchGate link.
Index entries for sequences related to polyominoes.

Cf. A000105, A130866, A246521, A368661, A368662, A368863 (fixed polyominoes).
Rows 3-9 are A368663, A368664, A368665, A368666, A368667, A368668, A368669.
Corresponding sequences for internal diffusion-limited aggregation: A368386, A368387.

A368392 a(n) is the numerator of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 17, 1, 1, 57, 5, 5, 5, 73, 5, 5, 73, 73, 5, 1, 5, 49321, 28165117, 5, 5, 169, 5, 123019, 63425, 28165117, 63425, 123019, 5, 49321, 5, 74999, 63425, 5, 58604629, 74999, 63425, 5, 5, 63425, 5, 74999, 5, 5, 63425, 74999, 63425, 5, 74999, 5000341, 32385, 5

Offset: 1

Pontus von Brömssen, Dec 22 2023

See A368386 for details.

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

			As an irregular triangle:
   1;
   1;
   1, 1;
   1, 4, 17, 1,  1;
  57, 5,  5, 5, 73, 5, 5, 73, 73, 5, 1, 5;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367675, A367764, A368000, A368386, A368387, A368393 (denominators), A368394, A368863 (external diffusion-limited aggregation).

a(n)/A368393(n) = (A368386(n)/A368387(n))/A335573(n+1).

A368393 a(n) is the denominator of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 2, 6, 6, 35, 35, 140, 35, 35, 1232, 1848, 1848, 1848, 3696, 1848, 1848, 3696, 3696, 1848, 7, 1848, 7386288, 3940584648, 38038, 38038, 5073, 38038, 7217188, 59034976, 3940584648, 59034976, 7217188, 38038, 7386288, 38038, 22138116, 59034976, 38038, 985146162, 22138116, 59034976, 38038, 38038, 59034976, 38038, 22138116, 38038, 38038, 59034976, 22138116, 59034976, 38038, 22138116, 177104928, 3689686, 38038

Offset: 1

Pontus von Brömssen, Dec 22 2023

See A368386 for details.

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

			As an irregular triangle:
     1;
     2;
     6,    6;
    35,   35,  140,   35,   35;
  1232, 1848, 1848, 1848, 3696, 1848, 1848, 3696, 3696, 1848, 7, 1848;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367676, A367765, A368001, A368386, A368387, A368392 (numerators), A368395, A368863 (external diffusion-limited aggregation).

A368392(n)/a(n) = (A368386(n)/A368387(n))/A335573(n+1).

A368865 Square array read by antidiagonals; the n-th row is the decimal expansion of the maximum probability that a particular fixed polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 0, 0, 0, 5, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 3, 8, 0, 0, 0, 0, 6, 1, 2, 0, 0, 0, 0, 5, 3, 8, 0, 0, 0, 0, 0, 6, 9, 3, 9, 0, 0, 0

Offset: 1

Pontus von Brömssen, Jan 08 2024

The n-th row is the decimal expansion of the maximum of the numbers corresponding to rows A130866(n-1)+1..A130866(n) of A368863.

It seems that the straight polyomino is the unique n-celled polyomino that has the maximum probability of appearing in a fixed orientation. If true, the n-th row here equals the A130866(n)-th row of A368863.

			Array begins:
  1.00000000000000000000... (1st row of A368863)
  0.50000000000000000000... (2nd row of A368863)
  0.21365625545581075649... (4th row of A368863)
  0.08139812221208792734... (9th row of A368863)
  0.02831865405554939733... (21st row of A368863)
  0.00913650301189504691... (56th row of A368863)
  ...

Index entries for sequences related to polyominoes.

Cf. A130866, A368394 (internal diffusion-limited aggregation), A368662 (free polyominoes), A368863, A368864 (minimum).

A368864 Square array read by antidiagonals; the n-th row is the decimal expansion of the minimum probability that a particular fixed polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 1, 7, 0, 0, 0, 0, 0, 7, 9, 9, 0, 0, 0, 0, 0, 1, 4, 3, 2, 0, 0, 0

Offset: 1

Pontus von Brömssen, Jan 08 2024

The n-th row is the decimal expansion of the minimum of the numbers corresponding to rows A130866(n-1)+1..A130866(n) of A368863.

It seems that the zig-zag polyomino is the unique n-celled polyomino that has the minimum probability of appearing in a fixed orientation.

			Array begins:
  1.00000000000000000000... (1st row of A368863)
  0.50000000000000000000... (2nd row of A368863)
  0.14317187227209462175... (3rd row of A368863)
  0.03794485956843370668... (8th row of A368863)
  0.00933365290110550590... (12th row of A368863)
  0.00216801081906196078... (42nd row of A368863)
  ...
The corresponding polyominoes for 1 <= n <= 6 are (all these are unique):
                                _        _ _
       _      _      _ _      _| |     _|  _|
  _   | |   _| |   _|  _|   _|  _|   _|  _|
 |_|  |_|  |_ _|  |_ _|    |_ _|    |_ _|

Index entries for sequences related to polyominoes.

Cf. A130866, A368661 (free polyominoes), A368863, A368865 (maximum).

cp's OEIS Frontend

A368660 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.

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A368392 a(n) is the numerator of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

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A368393 a(n) is the denominator of the probability that a particular one of the A335573(n+1) fixed polyominoes corresponding to the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

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A368865 Square array read by antidiagonals; the n-th row is the decimal expansion of the maximum probability that a particular fixed polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

Views

Author

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Examples

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Crossrefs

A368864 Square array read by antidiagonals; the n-th row is the decimal expansion of the minimum probability that a particular fixed polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs