A368002 -id:A368002 - cp's OEIS frontend

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.

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).

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.

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).

A368003 Denominator of the least 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, 2, 6, 28, 1001, 291291, 243373, 1634249695, 32872944939783505504120, 29741903410, 800946519853683297326312, 124513191408778015284779022326574057102628201495649607240, 4065704197026943714470537818579676

Offset: 1

Pontus von Brömssen, Dec 09 2023

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

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

Cf. A367997, A368000, A368001, A368002 (numerators), A368005.

cp's OEIS Frontend

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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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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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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A368003 Denominator of the least 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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