A368392 -id:A368392 - cp's OEIS frontend

A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

1, 1, 2, 1, 8, 4, 17, 4, 2, 57, 5, 5, 5, 73, 5, 5, 73, 73, 5, 1, 5, 49321, 28165117, 20, 20, 338, 20, 246038, 63425, 28165117, 63425, 123019, 20, 49321, 20, 149998, 63425, 20, 117209258, 74999, 63425, 10, 20, 63425, 20, 74999, 10, 10, 63425, 149998, 63425, 10, 149998, 5000341, 64770, 5

Offset: 1

Pontus von Brömssen, Dec 22 2023

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. a(n)/A368387(n) is the probability that, when the appropriate number of cells have been added, those cells 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, 17, 4,  2;
  57, 5,  5, 5, 73, 5, 5, 73, 73, 5, 1, 5;
  ...
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).
Persi Diaconis and William Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Semin. Mat. Univ. Politec. Torino, Vol. 49 (1991), No. 1, 95-119.
Gregory F. Lawler, Maury Bramson, and David Griffeath, Internal diffusion limited aggregation, The Annals of Probability 20 no. 4 (1992), 2117-2140.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367671, A367760, A367994, A368387 (denominators), A368388, A368390, A368392, A368393, A368660 (external diffusion-limited aggregation).

a(n)/A368387(n) = (A368392(n)/A368393(n))*A335573(n+1).

A368387 a(n) is the denominator of the probability that 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, 1, 3, 3, 35, 35, 35, 35, 35, 154, 462, 462, 231, 462, 231, 462, 924, 462, 462, 7, 924, 1846572, 492573081, 19019, 19019, 5073, 19019, 1804297, 7379372, 492573081, 7379372, 1804297, 19019, 1846572, 19019, 5534529, 7379372, 19019, 492573081, 5534529, 7379372, 19019, 19019, 7379372, 19019, 5534529, 19019, 19019, 14758744, 5534529, 7379372, 19019, 5534529, 44276232, 1844843, 19019

Offset: 1

Pontus von Brömssen, Dec 22 2023

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. A368386(n)/a(n) is the probability that, when the appropriate number of cells have been added, those cells 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;
   35,  35,  35,  35,  35;
  154, 462, 462, 231, 462, 231, 462, 924, 462, 462, 7, 924;
  ...
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).
Persi Diaconis and William Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Semin. Mat. Univ. Politec. Torino, Vol. 49 (1991), No. 1, 95-119.
Gregory F. Lawler, Maury Bramson, and David Griffeath, Internal diffusion limited aggregation, The Annals of Probability 20 no. 4 (1992), 2117-2140.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367672, A367761, A367995, A368386 (numerators), A368389, A368391, A368392, A368393, A368660 (external diffusion-limited aggregation).

A368386(n)/a(n) = (A368392(n)/A368393(n))*A335573(n+1).

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

Pontus von Brömssen, Dec 22 2023

See A368386 for details.

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

			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;
  ...

Index entries for sequences related to polyominoes.

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

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

A368863 Square array read by antidiagonals; the n-th row is the decimal expansion 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 diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 4, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 1, 3, 5, 0, 0, 0, 0, 7, 6, 3, 5, 0, 0, 0, 0, 1, 5, 3, 4, 5, 0, 0, 0, 0, 8, 6, 1, 6, 1, 3, 0, 0, 0, 0, 7, 2, 1, 2, 0, 7, 8, 0, 0, 0, 0, 2, 5, 7, 9, 7, 9, 1, 1, 0, 0, 0, 0, 2, 5, 4, 4, 5, 4, 3, 6, 1, 0, 0

Offset: 1

Pontus von Brömssen, Jan 08 2024

The n-th row is the decimal expansion of the number on the n-th row of A368660 divided by A335573(n+1). See A368660 for details.

Rows A130866(k-1)+1 to A130866(k) correspond to k-celled polyominoes, k >= 2.

			Array begins:
  1.00000000000000000000... (monomino)
  0.50000000000000000000... (domino)
  0.14317187227209462175... (L tromino)
  0.21365625545581075649... (I tromino)
  0.05331174468766310877... (L tetromino)
  0.05462942885357382723... (square tetromino)
  0.05107523273680265528... (T tetromino)
  0.03794485956843370668... (S tetromino)
  0.08139812221208792734... (I tetromino)
  0.01652391644265825925... (P pentomino)
  0.01709341200261444870... (V pentomino)
  0.00933365290110550590... (W pentomino)
  0.01825698429438352158... (L pentomino)
  0.01973313069852314774... (Y pentomino)
  0.01316184592639931744... (N pentomino)
  0.01069856796007681265... (U pentomino)
  0.02067501830899727807... (T pentomino)
  0.01358243200363682514... (F pentomino)
  0.01232428737930631004... (Z pentomino)
  0.01279646275569121440... (X pentomino)
  0.02831865405554939733... (I pentomino)
  ...

Pontus von Brömssen, Table of n, a(n) for n = 1..1596 (first 56 antidiagonals).
Index entries for sequences related to polyominoes.

Cf. A000105, A001168, A130866, A246521, A335573, A368660 (free polyominoes), A368864, A368865.

Corresponding sequences for internal diffusion-limited aggregation: A368392, A368393.

A368394 Numerator of the greatest probability that a particular fixed polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 1, 1, 17, 1, 58604629, 7301173188011, 115754318583755964857, 42019331987769250981907399, 8401384904285310565650785385525173372621364715976628525884130138767724737789789512541, 37312539934277875075756604487432403113653588096265391102288243043902545095467233603420824779574618387173667051527271

Offset: 1

Pontus von Brömssen, Dec 22 2023

a(n) is the numerator of the maximum of A368392/A368393 over the n-th row.

See A368386 for details.

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

Cf. A367677, A367766, A368004, A368386, A368390, A368392, A368393, A368395 (denominators), A368865 (external diffusion-limited aggregation).

A368395 Denominator of the greatest probability that a particular fixed polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 2, 6, 140, 7, 985146162, 190373377530336, 4103913140371011711744, 1900892466087624965855151720, 615876607410431086757183559738769319966704415344178702618301644074612533917929905792000, 4113793825406037267512938546695751842153799476671563574067367370457531066126881303220587185019966105893663857242112000

Offset: 1

Pontus von Brömssen, Dec 22 2023

a(n) is the denominator of the maximum of A368392/A368393 over the n-th row.

See A368386 for details.

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

Cf. A367678, A367767, A368005, A368386, A368391, A368392, A368393, A368394 (numerators), A368865 (external diffusion-limited aggregation).

cp's OEIS Frontend

A368386 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

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A368387 a(n) is the denominator of the probability that the free polyomino with binary code A246521(n+1) appears in internal diffusion-limited aggregation on the square lattice.

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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.

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Formula

A368863 Square array read by antidiagonals; the n-th row is the decimal expansion 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 diffusion-limited aggregation on the square lattice.

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A368394 Numerator of the greatest probability that a particular fixed polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

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Author

Keywords

Comments

Links

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A368395 Denominator of the greatest probability that a particular fixed polyomino with n cells appears in internal diffusion-limited aggregation on the square lattice.

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Author

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