A367762 -id:A367762 - cp's OEIS frontend

A367760 a(n) is the numerator 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.

1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 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

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, with probabilities proportional to the number of already existing squares with which the new square shares an edge. This seems to be the version described in Eden (1961). See A367671 for another 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;
  1, 1, 1, 1, 1;
  2, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 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 * 1/4), so the probability of obtaining the T-tetromino is 1/12 + 1/6 = 1/4 and a(7) = 1.

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, A367671, A367761 (denominators), A367762, A367764, A367765.

a(n)/A367761(n) = (A367764(n)/A367765(n))*A335573(n+1).

A367673 Numerator of the greatest probability that a particular free polyomino with n cells appears in the version of the Eden growth model described in A367671.

Original entry on oeis.org

1, 1, 2, 5, 253, 2657, 1839533, 11611594193, 119221101341, 3152002318138937, 2390156990671551019, 391219943696485871537172611, 1374972518894998705708792681, 21138479762006403022428257137861

Offset: 1

Pontus von Brömssen, Nov 26 2023

a(n) is the numerator of the maximum of A367671/A367672 over the n-th row.

			For 1 <= n <= 14, the following are the unique polyominoes that have the maximum probabilities for their respective sizes:
                    _      _      _
        _    _     | |    | |_   | |_ _
   _   | |  | |_   | |_   |   |  |    _|
  |_|  |_|  |_ _|  |_ _|  |_ _|  |_ _|
                                      _ _
     _          _          _ _      _|   |_
   _| |_ _    _| |_ _    _|   |_   |      _|
  |_     _|  |      _|  |      _|  |_ _  |
    |_ _|    |_ _ _|    |_ _ _|        |_|
                           _          _
     _ _      _ _ _       | |_      _| |_
   _|   |_   |     |_    _|   |_   |     |_
  |      _|  |      _|  |       |  |       |
  |_    |    |_    |    |_     _|  |_     _|
    |_ _|      |_ _|      |_ _|      |_ _|

Index entries for sequences related to polyominoes.

Cf. A367671, A367672, A367674 (denominators), A367677, A367762.

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).

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.

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).

cp's OEIS Frontend

A367760 a(n) is the numerator 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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A367673 Numerator of the greatest probability that a particular free polyomino with n cells appears in the version of the Eden growth model described in A367671.

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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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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.

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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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