A367998 -id:A367998 - 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).

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.

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.

A368004 Numerator of the greatest probability that a particular fixed 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, 4, 97, 2495, 98576101, 790070277194753299070819, 1697817285476742288131092, 301424494727669492958807965129775458632594691220000993251280413656197020195992465248816242330162

Offset: 1

Pontus von Brömssen, Dec 21 2023

a(n) is the numerator of the maximum of A368000/A368001 over the n-th row. See A368000 for details.

			For 1 <= n <= 13, 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).
                    _           _
        _    _     | |   _ _   | |_
   _   | |  | |_   | |  |   |  |   |
  |_|  |_|  |_ _|  |_|  |_ _|  |_ _|
   _ _      _ _    _ _      _ _ _
  |   |   _|   |  |   |_   |     |
  |   |  |    _|  |     |  |     |
  |_ _|  |_ _|    |_ _ _|  |_ _ _|
     _ _    _ _      _ _ _      _ _
   _|   |  |   |_   |     |   _|   |_
  |     |  |     |  |     |  |       |
  |    _|  |     |  |     |  |      _|
  |_ _|    |_ _ _|  |_ _ _|  |_ _ _|

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

Cf. A367677, A367766, A367998, A368000, A368001, A368002, A368005 (denominators).

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

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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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.

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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.

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A368004 Numerator of the greatest probability that a particular fixed 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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