A367995 -id:A367995 - cp's OEIS frontend

A367994 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears as the image of a simple random walk on the square lattice.

1, 1, 2, 1, 8, 4, 1, 4, 2, 388, 4, 4, 8, 64, 8, 4, 32, 64, 4, 1, 2, 3468, 76520, 4, 4, 2495, 4, 2102248, 1556, 76520, 1556, 1051124, 4, 3468, 4, 1194, 1556, 4, 1262762, 597, 1556, 2, 4, 1556, 4, 597, 2, 2, 778, 1194, 1556, 2, 1194, 2501, 1648, 1, 5270, 13652575732976, 13652575732976, 4468, 4468

Offset: 1

Pontus von Brömssen, Dec 08 2023

In a simple random walk on the square lattice, draw a unit square around each visited point. a(n)/A367995(n) is the probability that, when the appropriate number of distinct points have been visited, the drawn squares 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, 1, 4,  2;
  388, 4, 4, 8, 64, 8, 4, 32, 64, 4, 1, 2;
  ...
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).
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367671, A367760, A367995 (denominators), A367996, A367998, A368000, A368001, A368386.

a(n)/A367995(n) = (A368000(n)/A368001(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).

A368000 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 as the image of a simple random walk on the square lattice.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 97, 1, 1, 1, 8, 1, 1, 8, 8, 1, 1, 1, 867, 9565, 1, 1, 2495, 1, 262781, 389, 9565, 389, 262781, 1, 867, 1, 597, 389, 1, 631381, 597, 389, 1, 1, 389, 1, 597, 1, 1, 389, 597, 389, 1, 597, 2501, 412, 1, 2635, 1706571966622, 1706571966622, 1117, 1117

Offset: 1

Pontus von Brömssen, Dec 09 2023

In a simple random walk on the square lattice, draw a unit square around each visited point. a(n)/A368001(n) is the probability that, when the appropriate number of distinct points have been visited, the drawn squares form a particular one of the fixed polyominoes corresponding to 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;
   1, 1;
   1, 4, 1, 1, 1;
  97, 1, 1, 1, 8, 1, 1, 8, 8, 1, 1, 1;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367675, A367764, A367994, A367995, A368001 (denominators), A368002, A368004.

a(n)/A368001(n) = (A367994(n)/A367995(n))/A335573(n+1).

A368001 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 as the image of a simple random walk on the square lattice.

Original entry on oeis.org

1, 2, 6, 6, 21, 21, 28, 21, 21, 2002, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77, 89089, 785603, 286, 286, 48594, 286, 25924899, 194194, 785603, 194194, 25924899, 286, 89089, 286, 388388, 194194, 286, 51849798, 388388, 194194, 286, 286, 194194, 286, 388388, 286, 286, 194194, 388388, 194194, 286, 388388, 1165164, 291291, 286

Offset: 1

Pontus von Brömssen, Dec 09 2023

In a simple random walk on the square lattice, draw a unit square around each visited point. A368000(n)/a(n) is the probability that, when the appropriate number of distinct points have been visited, the drawn squares form a particular one of the fixed polyominoes corresponding to 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;
     2;
     6,  6;
    21, 21, 28, 21,   21;
  2002, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367676, A367765, A367994, A367995, A368000 (numerators), A368003, A368005.

A368000(n)/a(n) = (A367994(n)/A367995(n))/A335573(n+1).

A367996 Numerator of the least probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

Original entry on oeis.org

1, 1, 1, 2, 1, 1648, 10916, 227056, 17, 37138, 32596907, 203911047902268383, 61

Offset: 1

Pontus von Brömssen, Dec 08 2023

a(n) is the numerator of the minimum of A367994/A367995 over the n-th row. See A367994 for details.

Pontus von Brömssen, Illustration of all least probable polyominoes with at most 13 cells.
Index entries for sequences related to polyominoes.

Cf. A367994, A367995, A367997 (denominators), A367998, A368002.

A367998 Numerator of the greatest probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

Original entry on oeis.org

1, 1, 2, 8, 388, 2495, 13652575732976, 1580140554389506598141638, 2303282945504494379369753334706333784257298061180917309, 1116351824215919296474220471583292515147278170740521646743561595082143234790184233409933250330039986837258312677349601942095851

Offset: 1

Pontus von Brömssen, Dec 08 2023

a(n) is the numerator of the maximum of A367994/A367995 over the n-th row. See A367994 for details.

			For 1 <= n <= 13, the following are all polyominoes that have the maximum probabilities for their respective sizes. Except for n = 7, there is just one such polyomino.
                    _      _      _ _
        _    _     | |    | |_   |   |
   _   | |  | |_   | |_   |   |  |   |
  |_|  |_|  |_ _|  |_ _|  |_ _|  |_ _|
            _                 _ _
   _ _     | |_    _ _      _|   |
  |   |    |   |  |   |_   |    _|
  |   |_   |   |  |     |  |   |
  |_ _ _|  |_ _|  |_ _ _|  |_ _|
   _ _      _ _        _ _        _ _ _
  |   |    |   |_    _|   |_    _|     |
  |   |_   |     |  |       |  |      _|
  |     |  |     |  |    _ _|  |     |
  |_ _ _|  |_ _ _|  |_ _|      |_ _ _|

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

Cf. A367673, A367762, A367994, A367995, A367996, A367999 (denominators), A368004.

A367999 Denominator of the greatest probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

Original entry on oeis.org

1, 1, 3, 21, 1001, 24297, 154359847292651, 30341774437965821386991435, 92319517852923871319659686774769508009256168960677900730, 50934152340027691948241452572262612821964943639897156747372521002131242728356002575294796863242927131886444334117126282630281250

Offset: 1

Pontus von Brömssen, Dec 08 2023

a(n) is the denominator of the maximum of A367994/A367995 over the n-th row. See A367994 for details.

			See A367998.

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

Cf. A367674, A367763, A367994, A367995, A367997, A367998 (numerators), A368005.

A367997 Denominator of the least probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

Original entry on oeis.org

1, 1, 3, 21, 91, 291291, 17574557, 1433685253, 2819787, 14870951705, 124958680680282, 3525784478869018946300814, 27359333221

Offset: 1

Pontus von Brömssen, Dec 08 2023

a(n) is the denominator of the minimum of A367994/A367995 over the n-th row. See A367994 for details.

Pontus von Brömssen, Illustration of all least probable polyominoes with at most 13 cells.
Index entries for sequences related to polyominoes.

Cf. A367994, A367995, A367996 (numerators), A367999, A368003.

cp's OEIS Frontend

A367994 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears as the image of a simple random walk 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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A368000 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 as the image of a simple random walk on the square lattice.

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A368001 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 as the image of a simple random walk on the square lattice.

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Crossrefs

Formula

A367996 Numerator of the least probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

Views

Author

Keywords

Comments

Links

Crossrefs

A367998 Numerator of the greatest probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A367999 Denominator of the greatest probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A367997 Denominator of the least probability that a particular free polyomino with n cells appears as the image of a simple random walk on the square lattice.

Views

Author

Keywords

Comments

Links

Crossrefs