A367760 -id:A367760 - cp's OEIS frontend

A367764 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 in the Eden growth model on the square lattice (see A367760), when n square cells have been added.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 1, 23, 49, 1, 1, 53, 1, 107, 1, 49, 1, 107, 1, 23, 1, 1, 1, 1, 137, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 7, 1, 2797, 70037, 70037, 31, 31, 2797, 3517, 1, 41, 653, 49541, 1, 3517, 71, 67, 41, 899, 2797, 653, 1, 1, 1, 1, 653, 1, 1

Offset: 1

Pontus von Brömssen, Dec 02 2023

Apparently, the probabilities a(n)/A367765(n) are given in Eden (1958) for polyominoes up to 8 cells.

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, 1, 1, 1, 1;
  1, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 1;
  ...

Murray Eden, A probabilistic model for morphogenesis, in: Symposium on Information Theory in Biology (Gatlinburg 1956), pp. 359-370, Pergamon Press, New York, 1958.

Murray Eden, A two-dimensional growth process, in: 4th Berkeley Symposium on Mathematical Statistics and Probability (Berkeley 1960), vol. 4, pp. 223-239, University of California Press, Berkeley, 1961.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367675, A367760, A367761, A367765 (denominators), A367766.

a(n)/A367765(n) = (A367760(n)/A367761(n))/A335573(n+1).

A367765 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 in the Eden growth model on the square lattice (see A367760), when n square cells have been added.

Original entry on oeis.org

1, 2, 6, 6, 24, 6, 16, 24, 24, 20, 120, 120, 120, 480, 120, 120, 480, 480, 120, 40, 120, 1800, 4800, 720, 720, 1200, 720, 7200, 384, 4800, 384, 7200, 720, 1800, 720, 320, 384, 720, 7200, 320, 384, 720, 720, 384, 720, 320, 720, 720, 384, 320, 384, 720, 320, 1920, 1440, 720

Offset: 1

Pontus von Brömssen, Dec 02 2023

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;
  24,   6,  16,  24,  24;
  20, 120, 120, 120, 480, 120, 120, 480, 480, 120, 40, 120;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367676, A367760, A367761, A367764 (numerators), A367767.

A367764(n)/a(n) = (A367760(n)/A367761(n))/A335573(n+1).

A367762 Numerator of the greatest probability that a particular free polyomino with n cells appears in the Eden growth model (see A367760).

Original entry on oeis.org

1, 1, 2, 1, 2, 107, 70037, 813359, 1523168309, 1248684827, 26754412658849, 21916760758464961, 967387787846413936477, 271537923398149191011927

Offset: 1

Pontus von Brömssen, Dec 02 2023

a(n) is the numerator of the maximum of A367760/A367761 over the n-th row.

			For 1 <= n <= 14, the following are all polyominoes that have the maximum probabilities for their respective sizes. Except for n = 7, there is just one such polyomino.
                    _      _      _
        _    _     | |    | |_   | |_ _
   _   | |  | |_   | |_   |   |  |    _|
  |_|  |_|  |_ _|  |_ _|  |_ _|  |_ _|
            _                 _      _
   _ _     | |_    _ _      _| |_   | |_ _
  |   |    |   |  |   |_   |     |  |     |
  |   |_   |   |  |     |  |    _|  |     |
  |_ _ _|  |_ _|  |_ _ _|  |_ _|    |_ _ _|
   _ _        _ _      _ _ _      _ _ _
  |   |_    _|   |    |     |    |     |
  |     |  |     |_   |     |_   |     |_
  |     |  |      _|  |      _|  |       |
  |_ _ _|  |_ _ _|    |_ _ _|    |_ _ _ _|

Index entries for sequences related to polyominoes.

Cf. A367673, A367760, A367761, A367763 (denominators), A367766.

A367763 Denominator of the greatest probability that a particular free polyomino with n cells appears in the Eden growth model (see A367760).

Original entry on oeis.org

1, 1, 3, 3, 5, 900, 756000, 15876000, 42674688000, 56899584000, 1204450394112000, 2782280410398720000, 154249625952505036800000, 61082851877191994572800000

Offset: 1

Pontus von Brömssen, Dec 02 2023

a(n) is the denominator of the maximum of A367760/A367761 over the n-th row.

Index entries for sequences related to polyominoes.

Cf. A367674, A367760, A367761, A367762 (numerators), A367767.

A367766 Numerator of the greatest probability that a particular fixed polyomino with n cells appears in the Eden growth model (see A367760).

Original entry on oeis.org

1, 1, 1, 1, 1, 53, 49541, 813359, 59243701, 18007129909, 26754412658849, 922373240806979, 42709276740325681463, 5698447182281913432980459

Offset: 1

Pontus von Brömssen, Dec 02 2023

a(n) is the numerator of the maximum of A367764/A367765 over the n-th row.

			For 1 <= n <= 14, the following are all polyominoes, up to reflections and rotations, that have the maximum probabilities for their respective sizes. Except for n = 3, there is just one such polyomino (again, up to reflections and rotations).
                    _           _      _ _
        _    _     | |   _ _   | |_   |   |
   _   | |  | |_   | |  |   |  |   |  |   |
  |_|  |_|  |_ _|  |_|  |_ _|  |_ _|  |_ _|
                                _
   _ _      _ _      _ _ _    _| |_
  |   |_   |   |_   |     |  |     |
  |    _|  |     |  |     |  |     |
  |_ _|    |_ _ _|  |_ _ _|  |_ _ _|
   _ _      _ _ _      _ _      _ _ _
  |   |_   |     |   _|   |_   |     |_
  |     |  |     |  |       |  |       |
  |     |  |     |  |      _|  |      _|
  |_ _ _|  |_ _ _|  |_ _ _|    |_ _ _|

Index entries for sequences related to polyominoes.

Cf. A367677, A367760, A367762, A367764, A367765, A367767 (denominators).

A367767 Denominator of the greatest probability that a particular fixed polyomino with n cells appears in the Eden growth model (see A367760).

Original entry on oeis.org

1, 2, 6, 6, 20, 1200, 3024000, 63504000, 5334336000, 4779565056000, 9635603152896000, 404695332421632000, 44071321700715724800000, 7329942225263039348736000000

Offset: 1

Pontus von Brömssen, Dec 02 2023

a(n) is the denominator of the maximum of A367764/A367765 over the n-th row.

Index entries for sequences related to polyominoes.

Cf. A367678, A367760, A367763, A367764, A367765, A367766 (numerators).

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.

Original entry on oeis.org

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).

A367671 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in a version of the Eden growth model on the square lattice, when n square cells have been added.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 23, 4, 1, 253, 5, 1, 23, 713, 11, 5, 149, 157, 5, 23, 1, 3671, 286417, 16, 73, 289, 1, 2657, 103, 289, 15923, 19067, 1, 1661, 1, 10019, 16591, 1, 323, 193, 1661, 2, 169, 14603, 71, 853, 11, 23, 1037, 27151, 15923, 23, 529, 487, 14267, 1

Offset: 1

Pontus von Brömssen, Nov 26 2023

In the Eden growth model, there is a single initial unit square cell in the plane and more squares are added one at a time, selected randomly among those squares that share an edge with one of the already existing squares. In the version considered here, all such new squares have the same probability of being selected, whereas in Eden (1961) it appears that the probability is proportional to the number of already existing squares with which the new square shares an edge. See A367760 for the latter version.

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;
    5, 2, 23,  4,   1;
  253, 5,  1, 23, 713, 11, 5, 149, 157, 5, 23, 1;
  ...
For n = 7, the T-tetromino has binary code A246521(n+1) = 27. It can be obtained either via the straight tromino (probability 1/3 * 1/4) or via the L-tromino (probability 2/3 * 2/7), so the probability of obtaining the T-tetromino is 1/12 + 4/21 = 23/84 and a(7) = 23.

Murray Eden, A two-dimensional growth process, in: 4th Berkeley Symposium on Mathematical Statistics and Probability (Berkeley 1960), vol. 4, pp. 223-239, University of California Press, Berkeley, 1961.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367672 (denominators), A367673, A367675, A367676, A367760.

a(n)/A367672(n) = (A367675(n)/A367676(n))*A335573(n+1).

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.

Original entry on oeis.org

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).

A367761 a(n) is the denominator of the probability that the free polyomino with binary code A246521(n+1) appears in the Eden growth model on the square lattice, when n square cells have been added.

Original entry on oeis.org

1, 1, 3, 3, 3, 6, 4, 6, 12, 5, 30, 30, 15, 60, 15, 30, 120, 60, 30, 40, 60, 450, 600, 90, 90, 600, 90, 900, 48, 600, 48, 1800, 90, 450, 90, 40, 48, 90, 1800, 80, 48, 180, 90, 48, 90, 80, 180, 180, 96, 40, 48, 180, 40, 480, 360, 360, 151200, 756000, 756000, 10080, 10080, 151200, 151200, 630, 10080

Offset: 1

Pontus von Brömssen, Dec 02 2023

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;
  3,  6,  4,  6, 12;
  5, 30, 30, 15, 60, 15, 30, 120, 60, 30, 40, 60;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367672, A367760 (numerators), A367763, A367764, A367765.

A367760(n)/a(n) = (A367764(n)/A367765(n))*A335573(n+1).

cp's OEIS Frontend

A367764 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 in the Eden growth model on the square lattice (see A367760), when n square cells have been added.

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A367765 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 in the Eden growth model on the square lattice (see A367760), when n square cells have been added.

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A367762 Numerator of the greatest probability that a particular free polyomino with n cells appears in the Eden growth model (see A367760).

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A367763 Denominator of the greatest probability that a particular free polyomino with n cells appears in the Eden growth model (see A367760).

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A367766 Numerator of the greatest probability that a particular fixed polyomino with n cells appears in the Eden growth model (see A367760).

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A367767 Denominator of the greatest probability that a particular fixed polyomino with n cells appears in the Eden growth model (see A367760).

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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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A367671 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in a version of the Eden growth model on the square lattice, when n square cells have been added.

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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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A367761 a(n) is the denominator of the probability that the free polyomino with binary code A246521(n+1) appears in the Eden growth model on the square lattice, when n square cells have been added.

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