A368662 -id:A368662 - 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).

A368665 Decimal expansion of the probability that the L tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

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

Offset: 0

Pontus von Brömssen, Jan 04 2024

See A368660 for details.

			0.426493957501304870185842561289148822006185205656733654816895...

5th row of A368660.
4th row of A368662.
Cf. A368666, A368667, A368668, A368669.

Equals (4644864 - 9252864*Pi + 7592128*Pi^2 - 3288992*Pi^3 + 794310*Pi^4 - 101490*Pi^5 + 5364*Pi^6)/(19267584 - 39854080*Pi + 33814528*Pi^2 - 15105856*Pi^3 + 3754992*Pi^4 - 493215*Pi^5 + 26775*Pi^6).

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

A368390 Numerator of the greatest 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, 2, 17, 57, 117209258, 363356390591, 292511604691740220504677006975130493521391, 2806200956345391684353279299766025388803856326541039774177, 8401384904285310565650785385525173372621364715976628525884130138767724737789789512541

Offset: 1

Pontus von Brömssen, Dec 22 2023

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

			For 1 <= n <= 13, the following are the unique polyominoes that have the maximum probabilities for their respective sizes:
                    _      _        _
        _    _     | |_   | |_    _| |_
   _   | |  | |_   |  _|  |   |  |_    |
  |_|  |_|  |_ _|  |_|    |_ _|    |_ _|
     _          _          _ _      _ _ _
   _| |_ _    _| |_ _    _|   |_   |     |_
  |_     _|  |      _|  |      _|  |      _|
    |_ _|    |_ _ _|    |_ _ _|    |_ _ _|
                           _
              _ _ _      _| |_
   _ _ _     |     |_   |     |_
  |     |_   |       |  |       |
  |       |  |_   _ _|  |_   _ _|
  |_ _ _ _|    |_|        |_|

Pontus von Brömssen, Table of n, a(n) for n = 1..13
Index entries for sequences related to polyominoes.

Cf. A367673, A367762, A367998, A368386, A368387, A368388, A368391 (denominators), A368394, A368662 (external diffusion-limited aggregation).

A368391 Denominator of the greatest 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, 154, 492573081, 2263639868336, 2851008421256134796827810712799257125795200, 23733667150026775615685852367392733768076432660227006271200, 153969151852607771689295889934692329991676103836044675654575411018653133479482476448000

Offset: 1

Pontus von Brömssen, Dec 22 2023

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

			See A368390.

Pontus von Brömssen, Table of n, a(n) for n = 1..13
Index entries for sequences related to polyominoes.

Cf. A367674, A367763, A367999, A368386, A368387, A368389, A368390 (numerators), A368395, A368662 (external diffusion-limited aggregation).

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).

A368663 Decimal expansion of the probability that the L tromino appears when the 3rd cell is added in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

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

Offset: 0

Pontus von Brömssen, Jan 04 2024

See A368660 for details.

			0.572687489088378487018245865067925713098269095362316686554472...

Index entries for sequences related to polyominoes.

3rd row of A368660.
3rd row of A368662.
Cf. A368664.

First[RealDigits[(20 - 6*Pi)/(24 - 7*Pi), 10, 100]] (* Paolo Xausa, Nov 12 2024 *)

Equals (20-6*Pi)/(24-7*Pi) = 1 - A368664.

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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A368665 Decimal expansion of the probability that the L tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

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A368390 Numerator of the greatest probability that a particular free polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

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A368391 Denominator of the greatest probability that a particular free polyomino with n cells 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.

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A368663 Decimal expansion of the probability that the L tromino appears when the 3rd cell is added in diffusion-limited aggregation on the square lattice.

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