A368660 -id:A368660 - cp's OEIS frontend

A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

1, 1, 2, 1, 8, 4, 17, 4, 2, 57, 5, 5, 5, 73, 5, 5, 73, 73, 5, 1, 5, 49321, 28165117, 20, 20, 338, 20, 246038, 63425, 28165117, 63425, 123019, 20, 49321, 20, 149998, 63425, 20, 117209258, 74999, 63425, 10, 20, 63425, 20, 74999, 10, 10, 63425, 149998, 63425, 10, 149998, 5000341, 64770, 5

Offset: 1

Pontus von Brömssen, Dec 22 2023

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. a(n)/A368387(n) is the probability that, when the appropriate number of cells have been added, those cells form the free polyomino with binary code A246521(n+1).

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

			As an irregular triangle:
   1;
   1;
   2, 1;
   8, 4, 17, 4,  2;
  57, 5,  5, 5, 73, 5, 5, 73, 73, 5, 1, 5;
  ...
There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
For three squares, the probability for an L (or right) tromino (whose binary code is 7 = A246521(4)) is 2/3, so a(3) = 2. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 1.

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).
Persi Diaconis and William Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Semin. Mat. Univ. Politec. Torino, Vol. 49 (1991), No. 1, 95-119.
Gregory F. Lawler, Maury Bramson, and David Griffeath, Internal diffusion limited aggregation, The Annals of Probability 20 no. 4 (1992), 2117-2140.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367671, A367760, A367994, A368387 (denominators), A368388, A368390, A368392, A368393, A368660 (external diffusion-limited aggregation).

a(n)/A368387(n) = (A368392(n)/A368393(n))*A335573(n+1).

A368387 a(n) is the denominator of the probability that 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, 3, 3, 35, 35, 35, 35, 35, 154, 462, 462, 231, 462, 231, 462, 924, 462, 462, 7, 924, 1846572, 492573081, 19019, 19019, 5073, 19019, 1804297, 7379372, 492573081, 7379372, 1804297, 19019, 1846572, 19019, 5534529, 7379372, 19019, 492573081, 5534529, 7379372, 19019, 19019, 7379372, 19019, 5534529, 19019, 19019, 14758744, 5534529, 7379372, 19019, 5534529, 44276232, 1844843, 19019

Offset: 1

Pontus von Brömssen, Dec 22 2023

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. A368386(n)/a(n) is the probability that, when the appropriate number of cells have been added, those cells form the free polyomino with binary code A246521(n+1).

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

			As an irregular triangle:
    1;
    1;
    3,   3;
   35,  35,  35,  35,  35;
  154, 462, 462, 231, 462, 231, 462, 924, 462, 462, 7, 924;
  ...
There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
For three squares, the probability for an L (or right) tromino (whose binary code is 7 = A246521(4)) is 2/3, so a(3) = 3. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 3.

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).
Persi Diaconis and William Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Semin. Mat. Univ. Politec. Torino, Vol. 49 (1991), No. 1, 95-119.
Gregory F. Lawler, Maury Bramson, and David Griffeath, Internal diffusion limited aggregation, The Annals of Probability 20 no. 4 (1992), 2117-2140.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367672, A367761, A367995, A368386 (numerators), A368389, A368391, A368392, A368393, A368660 (external diffusion-limited aggregation).

A368386(n)/a(n) = (A368392(n)/A368393(n))*A335573(n+1).

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).

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.

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.

A368667 Decimal expansion of the probability that the T tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

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

Offset: 0

Pontus von Brömssen, Jan 04 2024

See A368660 for details.

			0.204300930947210621152745074614757706834953524705637911204238...

7th row of A368660.

Cf. A368665, A368666, A368668, A368669.

Equals (286720 - 520576*Pi + 365408*Pi^2 - 125052*Pi^3 + 20982*Pi^4 - 1386*Pi^5)/(1204224 - 2114560*Pi + 1452608*Pi^2 - 490176*Pi^3 + 81507*Pi^4 - 5355*Pi^5).

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

A368668 Decimal expansion of the probability that the skew tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

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

Offset: 0

Pontus von Brömssen, Jan 04 2024

See A368660 for details.

			0.151779438273734826738989340932032862591131082461170163290667...

8th row of A368660.

Cf. A368665, A368666, A368667, A368669.

Equals (3200 - 5560*Pi + 3370*Pi^2 - 867*Pi^3 + 81*Pi^4)/(21504 - 32000*Pi + 17368*Pi^2 - 4101*Pi^3 + 357*Pi^4).

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

A368669 Decimal expansion of the probability that the straight tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

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

Offset: 0

Pontus von Brömssen, Jan 04 2024

See A368660 for details.

			0.162796244424175854682512001024548670877500478908211364328916...

9th row of A368660.

Cf. A368665, A368666, A368667, A368668.

First[RealDigits[2*(Pi - 4)^2*(3*Pi - 10)/((5*Pi - 16)*(7*Pi - 24)*(15*Pi - 56)), 10, 100]] (* Paolo Xausa, Nov 12 2024 *)

Equals (320 - 256*Pi + 68*Pi^2 - 6*Pi^3)/(21504 - 18752*Pi + 5440*Pi^2 - 525*Pi^3).

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

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.

cp's OEIS Frontend

A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

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A368387 a(n) is the denominator of the probability that the free polyomino with binary code A246521(n+1) appears in internal 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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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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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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A368667 Decimal expansion of the probability that the T tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

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A368668 Decimal expansion of the probability that the skew tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

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A368669 Decimal expansion of the probability that the straight tetromino appears when the 4th cell is added in diffusion-limited aggregation on the square lattice.

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