cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

Pontus von Brömssen, Dec 08 2023

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A000105, A246521, A335573, A367671, A367760, A367995 (denominators), A367996, A367998, A368000, A368001, A368386.

Formula

a(n)/A367995(n) = (A368000(n)/A368001(n))*A335573(n+1).

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.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Dec 08 2023

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Formula

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

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.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Dec 09 2023

Keywords

Comments

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.

Examples

			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;
  ...

Links

Crossrefs

Cf. A000105, A246521, A335573, A367675, A367764, A367994, A367995, A368001 (denominators), A368002, A368004.

Formula

a(n)/A368001(n) = (A367994(n)/A367995(n))/A335573(n+1).

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

Views

Author

Pontus von Brömssen, Dec 21 2023

Keywords

Comments

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

Examples

			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).
                    _           _
        _    _     | |   _ _   | |_
   _   | |  | |_   | |  |   |  |   |
  |_|  |_|  |_ _|  |_|  |_ _|  |_ _|
   _ _      _ _    _ _      _ _ _
  |   |   _|   |  |   |_   |     |
  |   |  |    _|  |     |  |     |
  |_ _|  |_ _|    |_ _ _|  |_ _ _|
     _ _    _ _      _ _ _      _ _
   _|   |  |   |_   |     |   _|   |_
  |     |  |     |  |     |  |       |
  |    _|  |     |  |     |  |      _|
  |_ _|    |_ _ _|  |_ _ _|  |_ _ _|

Links

Crossrefs

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

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

Views

Author

Pontus von Brömssen, Dec 21 2023

Keywords

Comments

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

Examples

			See A368004.

Links

Crossrefs

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

A368393 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 internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 2, 6, 6, 35, 35, 140, 35, 35, 1232, 1848, 1848, 1848, 3696, 1848, 1848, 3696, 3696, 1848, 7, 1848, 7386288, 3940584648, 38038, 38038, 5073, 38038, 7217188, 59034976, 3940584648, 59034976, 7217188, 38038, 7386288, 38038, 22138116, 59034976, 38038, 985146162, 22138116, 59034976, 38038, 38038, 59034976, 38038, 22138116, 38038, 38038, 59034976, 22138116, 59034976, 38038, 22138116, 177104928, 3689686, 38038
Offset: 1

Views

Author

Pontus von Brömssen, Dec 22 2023

Keywords

Comments

See A368386 for details.
Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

Examples

			As an irregular triangle:
     1;
     2;
     6,    6;
    35,   35,  140,   35,   35;
  1232, 1848, 1848, 1848, 3696, 1848, 1848, 3696, 3696, 1848, 7, 1848;
  ...

Links

Crossrefs

Cf. A000105, A246521, A335573, A367676, A367765, A368001, A368386, A368387, A368392 (numerators), A368395, A368863 (external diffusion-limited aggregation).

Formula

A368392(n)/a(n) = (A368386(n)/A368387(n))/A335573(n+1).

A368002 Numerator 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, 1, 1, 1, 8, 412, 54, 59309, 176087633490076859, 18569, 81059275440235943, 1615066814060816060766626689229243976663152060069, 5644184595206308867273871
Offset: 1

Views

Author

Pontus von Brömssen, Dec 09 2023

Keywords

Comments

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

Links

Crossrefs

Cf. A367996, A368000, A368001, A368003 (denominators), A368004.

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

Views

Author

Pontus von Brömssen, Dec 09 2023

Keywords

Comments

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

Links

Crossrefs

Cf. A367997, A368000, A368001, A368002 (numerators), A368005.
Showing 1-8 of 8 results.

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