A130866 -id:A130866 - 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.

A368661 Square array read by antidiagonals; the n-th row is the decimal expansion of the minimum probability that a particular free polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

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

Offset: 1

Pontus von Brömssen, Jan 04 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 A368660. See A368660 for details.

			Array begins:
  1.00000000000000000000... (1st row of A368660)
  1.00000000000000000000... (2nd row of A368660)
  0.42731251091162151298... (4th row of A368660)
  0.05462942885357382723... (6th row of A368660)
  0.01279646275569121440... (20th row of A368660)
  0.00867204327624784314... (42nd row of A368660)
  ...
The corresponding polyominoes for 1 <= n <= 6 are (all these are unique):
            _             _          _ _
       _   | |   _ _    _| |_      _|  _|
  _   | |  | |  |   |  |_   _|   _|  _|
 |_|  |_|  |_|  |_ _|    |_|    |_ _|

Index entries for sequences related to polyominoes.

Cf. A130866, A368388 (internal diffusion-limited aggregation), A368660, A368662 (maximum), A368664 (row 3), A368666 (row 4), A368864 (fixed polyominoes).

A368662 Square array read by antidiagonals; the n-th row is the decimal expansion of the maximum probability that a particular free polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 7, 4, 0, 0, 0, 2, 2, 1, 0, 0, 0, 6, 6, 5, 0, 0, 0, 0, 8, 4, 7, 6, 0, 0

Offset: 1

Pontus von Brömssen, Jan 04 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 A368660. See A368660 for details.

			Array begins:
  1.00000000000000000000... (1st row of A368660)
  1.00000000000000000000... (2nd row of A368660)
  0.57268748908837848701... (3rd row of A368660)
  0.42649395750130487018... (5th row of A368660)
  0.15786504558818518196... (14th row of A368660)
  0.06192086165513502549... (36th row of A368660)
  ...
The corresponding polyominoes for 1 <= n <= 6 are (all these are unique):
                    _                   _
        _    _     | |       _         | |
   _   | |  | |_   | |_    _| |_ _    _| |_ _
  |_|  |_|  |_ _|  |_ _|  |_ _ _ _|  |_ _ _ _|

Index entries for sequences related to polyominoes.

Cf. A130866, A368390 (internal diffusion-limited aggregation), A368660, A368661 (minimum), A368663 (row 3), A368665 (row 4), A368865 (fixed polyominoes).

A368863 Square array read by antidiagonals; the n-th row is the decimal expansion 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 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, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 1, 3, 5, 0, 0, 0, 0, 7, 6, 3, 5, 0, 0, 0, 0, 1, 5, 3, 4, 5, 0, 0, 0, 0, 8, 6, 1, 6, 1, 3, 0, 0, 0, 0, 7, 2, 1, 2, 0, 7, 8, 0, 0, 0, 0, 2, 5, 7, 9, 7, 9, 1, 1, 0, 0, 0, 0, 2, 5, 4, 4, 5, 4, 3, 6, 1, 0, 0

Offset: 1

Pontus von Brömssen, Jan 08 2024

The n-th row is the decimal expansion of the number on the n-th row of A368660 divided by A335573(n+1). See A368660 for details.

Rows A130866(k-1)+1 to A130866(k) correspond to k-celled polyominoes, k >= 2.

			Array begins:
  1.00000000000000000000... (monomino)
  0.50000000000000000000... (domino)
  0.14317187227209462175... (L tromino)
  0.21365625545581075649... (I tromino)
  0.05331174468766310877... (L tetromino)
  0.05462942885357382723... (square tetromino)
  0.05107523273680265528... (T tetromino)
  0.03794485956843370668... (S tetromino)
  0.08139812221208792734... (I tetromino)
  0.01652391644265825925... (P pentomino)
  0.01709341200261444870... (V pentomino)
  0.00933365290110550590... (W pentomino)
  0.01825698429438352158... (L pentomino)
  0.01973313069852314774... (Y pentomino)
  0.01316184592639931744... (N pentomino)
  0.01069856796007681265... (U pentomino)
  0.02067501830899727807... (T pentomino)
  0.01358243200363682514... (F pentomino)
  0.01232428737930631004... (Z pentomino)
  0.01279646275569121440... (X pentomino)
  0.02831865405554939733... (I pentomino)
  ...

Pontus von Brömssen, Table of n, a(n) for n = 1..1596 (first 56 antidiagonals).
Index entries for sequences related to polyominoes.

Cf. A000105, A001168, A130866, A246521, A335573, A368660 (free polyominoes), A368864, A368865.

Corresponding sequences for internal diffusion-limited aggregation: A368392, A368393.

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).

A173271 Partial sums of A000105.

Original entry on oeis.org

1, 2, 3, 5, 10, 22, 57, 165, 534, 1819, 6474, 23547, 87147, 325738, 1227709, 4654285, 17733540, 67841449, 260463501, 1003087733, 3873759683, 14996820361, 58188678049, 226235685777, 881235386180, 3438462430944, 13437551253019

Offset: 0

Jonathan Vos Post, Feb 14 2010

nonn

John Mason, Table of n, a(n) for n = 0..50 (terms 0..48 from Jean-François Alcover).

Cf. A000105, A130866.

Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]] // Accumulate (* Jean-François Alcover, Aug 17 2022 *)

a(n) = A130866(n) + 1. - John Mason, Feb 21 2023

A130622 Number of polyominoes with perimeter at most 2n.

Original entry on oeis.org

0, 1, 2, 5, 11, 36, 122, 538, 2526, 13166, 71153, 400109, 2300430, 13504780, 80547558, 487327904

Offset: 1

Tanya Khovanova, Aug 10 2007

The perimeter of a polyomino is always even.
a(n) is partial sums of A057730.
a(n+1) >= A130866(n).
Is there another prime term beyond {2, 5, 11}?

Cf. A131482 (number of n-celled polyominoes with perimeter 2n+2).
Cf. A057730 (number of polyominoes (A000105) with perimeter 2n).
Cf. A130866 (number of polyominoes with at most n cells).

Offset corrected by John Mason, Jan 16 2023

a(9)-a(16) from John Mason, Jan 16 2023

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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A368661 Square array read by antidiagonals; the n-th row is the decimal expansion of the minimum probability that a particular free polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

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A368662 Square array read by antidiagonals; the n-th row is the decimal expansion of the maximum probability that a particular free polyomino with n cells appears in diffusion-limited aggregation on the square lattice.

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A368863 Square array read by antidiagonals; the n-th row is the decimal expansion 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 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.

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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.

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A173271 Partial sums of A000105.

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A130622 Number of polyominoes with perimeter at most 2n.

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