A367999 -id:A367999 - cp's OEIS frontend

A367995 a(n) is the denominator 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, 3, 3, 21, 21, 7, 21, 21, 1001, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77, 89089, 785603, 143, 143, 24297, 143, 25924899, 97097, 785603, 97097, 25924899, 143, 89089, 143, 97097, 97097, 143, 25924899, 97097, 97097, 143, 143, 97097, 143, 97097, 143, 143, 97097, 97097, 97097, 143, 97097, 291291, 291291, 143

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. A367994(n)/a(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;
     3,  3;
    21, 21,  7, 21,   21;
  1001, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77;
  ...
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).
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367672, A367761, A367994 (numerators), A367997, A367999, A368000, A368001.

A367994(n)/a(n) = (A368000(n)/A368001(n))*A335573(n+1).

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.

A368005 Denominator 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, 2, 6, 21, 2002, 48594, 6786340869, 60683548875931642773982870, 148748699073930002409397035, 98235230940726955523493708384725766221632599616516980478369143134298250712431827891413914221729825

Offset: 1

Pontus von Brömssen, Dec 21 2023

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

			See A368004.

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

Cf. A367678, A367767, A367999, A368000, A368001, A368003, A368004 (numerators).

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.

A368391 Denominator 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, 3, 35, 154, 492573081, 2263639868336, 2851008421256134796827810712799257125795200, 23733667150026775615685852367392733768076432660227006271200, 153969151852607771689295889934692329991676103836044675654575411018653133479482476448000

Offset: 1

Pontus von Brömssen, Dec 22 2023

a(n) is the denominator of the maximum of A368386/A368387 over the n-th row. See A368386 for details.

			See A368390.

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

Cf. A367674, A367763, A367999, A368386, A368387, A368389, A368390 (numerators), A368395, A368662 (external diffusion-limited aggregation).

cp's OEIS Frontend

A367995 a(n) is the denominator 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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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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A368005 Denominator 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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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.

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A368391 Denominator 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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