A368661 -id:A368661 - 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.

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).

A368666 Decimal expansion of the probability that the square tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

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

Offset: 0

Pontus von Brömssen, Jan 04 2024

See A368660 for details.

			0.054629428853573827239911022139511937690229708268246906359282...

6th row of A368660.
4th row of A368661.
Cf. A368665, A368667, A368668, A368669.

Equals (7680 - 10744*Pi + 5572*Pi^2 - 1272*Pi^3 + 108*Pi^4)/(21504 - 32000*Pi + 17368*Pi^2 - 4101*Pi^3 + 357*Pi^4).

A368665 + A368666 + A368667 + A368668 + A368669 = 1.

A368388 Numerator of the least probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 1, 1, 2, 5, 5, 101, 70, 1583, 7, 27877, 22, 537851329

Offset: 1

Pontus von Brömssen, Dec 22 2023

a(n) is the numerator of the minimum of A368386/A368387 over the n-th row. See A368386 for details.

For n <= 13, the straight polyomino has the least probability of appearing among all n-celled polyominoes, and it seems likely that this is true for all n.

Index entries for sequences related to polyominoes.

Cf. A367996, A368386, A368387, A368389 (denominators), A368390, A368661 (external diffusion-limited aggregation).

A368389 Denominator of the least probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 1, 3, 35, 924, 19019, 14774760, 767191139, 2455848787392, 2993165027255, 6240848877339043968, 5018882339663051609, 238246203631345609763405422080

Offset: 1

Pontus von Brömssen, Dec 22 2023

a(n) is the denominator of the minimum of A368386/A368387 over the n-th row. See A368386 for details.

For n <= 13, the straight polyomino has the least probability of appearing among all n-celled polyominoes, and it seems likely that this is true for all n.

Index entries for sequences related to polyominoes.

Cf. A367997, A368386, A368387, A368388 (numerators), A368391, A368661 (external diffusion-limited aggregation).

A368664 Decimal expansion of the probability that the straight tromino appears when the 3rd cell is added in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

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

Offset: 0

Pontus von Brömssen, Jan 04 2024

See A368660 for details.

			0.427312510911621512981754134932074286901730904637683313445527...

Index entries for sequences related to polyominoes.

4th row of A368660.
3rd row of A368661.
Cf. A368663.

First[RealDigits[(Pi - 4)/(7*Pi - 24), 10, 100]] (* Paolo Xausa, Nov 12 2024 *)

Equals (4-Pi)/(24-7*Pi) = 1 - A368663.

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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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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A368666 Decimal expansion of the probability that the square tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

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A368388 Numerator of the least probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

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A368389 Denominator of the least probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

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A368664 Decimal expansion of the probability that the straight tromino appears when the 3rd cell is added 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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