A335573 -id:A335573 - cp's OEIS frontend

A367675 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 in the version of the Eden growth model described in A367671 when n square cells have been added.

1, 1, 1, 1, 5, 2, 23, 1, 1, 253, 5, 1, 23, 713, 11, 5, 149, 157, 5, 23, 1, 3671, 286417, 2, 73, 289, 1, 2657, 103, 289, 15923, 19067, 1, 1661, 1, 10019, 16591, 1, 323, 193, 1661, 1, 169, 14603, 71, 853, 11, 23, 1037, 27151, 15923, 23, 529, 487, 14267, 1

Offset: 1

Pontus von Brömssen, Nov 26 2023

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;
    5, 2, 23,  1,   1;
  253, 5,  1, 23, 713, 11, 5, 149, 157, 5, 23, 1;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367671, A367672, A367676 (denominators), A367677, A367764.

a(n)/A367676(n) = (A367671(n)/A367672(n))/A335573(n+1).

A367676 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 the version of the Eden growth model described in A367671 when n square cells have been added.

Original entry on oeis.org

1, 2, 6, 6, 112, 21, 336, 21, 24, 8064, 504, 84, 2520, 40320, 1008, 504, 8064, 8064, 504, 672, 120, 399168, 39916800, 1155, 30240, 18144, 528, 241920, 26880, 36288, 4435200, 1814400, 480, 181440, 480, 2217600, 3991680, 528, 20736, 36288, 362880, 378, 110880, 4435200, 36960, 201600, 5040, 13860, 295680, 5702400, 4435200, 13860, 103680, 50400, 1814400, 720

Offset: 1

Pontus von Brömssen, Nov 26 2023

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

Terms on the n-th row are (2*n-1)-smooth.

			As an irregular triangle:
     1;
     2;
     6,   6;
   112,  21, 336,   21,    24;
  8064, 504,  84, 2520, 40320, 1008, 504, 8064, 8064, 504, 672, 120;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367671, A367672, A367675 (numerators), A367678, A367765.

A367675(n)/a(n) = (A367671(n)/A367672(n))/A335573(n+1).

A367764 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 in the Eden growth model on the square lattice (see A367760), when n square cells have been added.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 1, 23, 49, 1, 1, 53, 1, 107, 1, 49, 1, 107, 1, 23, 1, 1, 1, 1, 137, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 7, 1, 2797, 70037, 70037, 31, 31, 2797, 3517, 1, 41, 653, 49541, 1, 3517, 71, 67, 41, 899, 2797, 653, 1, 1, 1, 1, 653, 1, 1

Offset: 1

Pontus von Brömssen, Dec 02 2023

Apparently, the probabilities a(n)/A367765(n) are given in Eden (1958) for polyominoes up to 8 cells.

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, 1, 1, 1, 1;
  1, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 1;
  ...

Murray Eden, A probabilistic model for morphogenesis, in: Symposium on Information Theory in Biology (Gatlinburg 1956), pp. 359-370, Pergamon Press, New York, 1958.

Murray Eden, A two-dimensional growth process, in: 4th Berkeley Symposium on Mathematical Statistics and Probability (Berkeley 1960), vol. 4, pp. 223-239, University of California Press, Berkeley, 1961.
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367675, A367760, A367761, A367765 (denominators), A367766.

a(n)/A367765(n) = (A367760(n)/A367761(n))/A335573(n+1).

A367765 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 the Eden growth model on the square lattice (see A367760), when n square cells have been added.

Original entry on oeis.org

1, 2, 6, 6, 24, 6, 16, 24, 24, 20, 120, 120, 120, 480, 120, 120, 480, 480, 120, 40, 120, 1800, 4800, 720, 720, 1200, 720, 7200, 384, 4800, 384, 7200, 720, 1800, 720, 320, 384, 720, 7200, 320, 384, 720, 720, 384, 720, 320, 720, 720, 384, 320, 384, 720, 320, 1920, 1440, 720

Offset: 1

Pontus von Brömssen, Dec 02 2023

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;
  24,   6,  16,  24,  24;
  20, 120, 120, 120, 480, 120, 120, 480, 480, 120, 40, 120;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367676, A367760, A367761, A367764 (numerators), A367767.

A367764(n)/a(n) = (A367760(n)/A367761(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

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).

A368001 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 as the image of a simple random walk on the square lattice.

Original entry on oeis.org

1, 2, 6, 6, 21, 21, 28, 21, 21, 2002, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77, 89089, 785603, 286, 286, 48594, 286, 25924899, 194194, 785603, 194194, 25924899, 286, 89089, 286, 388388, 194194, 286, 51849798, 388388, 194194, 286, 286, 194194, 286, 388388, 286, 286, 194194, 388388, 194194, 286, 388388, 1165164, 291291, 286

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. A368000(n)/a(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;
     2;
     6,  6;
    21, 21, 28, 21,   21;
  2002, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367676, A367765, A367994, A367995, A368000 (numerators), A368003, A368005.

A368000(n)/a(n) = (A367994(n)/A367995(n))/A335573(n+1).

A368392 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 in internal diffusion-limited aggregation on the square lattice.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 17, 1, 1, 57, 5, 5, 5, 73, 5, 5, 73, 73, 5, 1, 5, 49321, 28165117, 5, 5, 169, 5, 123019, 63425, 28165117, 63425, 123019, 5, 49321, 5, 74999, 63425, 5, 58604629, 74999, 63425, 5, 5, 63425, 5, 74999, 5, 5, 63425, 74999, 63425, 5, 74999, 5000341, 32385, 5

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;
   1;
   1, 1;
   1, 4, 17, 1,  1;
  57, 5,  5, 5, 73, 5, 5, 73, 73, 5, 1, 5;
  ...

Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A335573, A367675, A367764, A368000, A368386, A368387, A368393 (denominators), A368394, A368863 (external diffusion-limited aggregation).

a(n)/A368393(n) = (A368386(n)/A368387(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.

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

A367675 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 in the version of the Eden growth model described in A367671 when n square cells have been added.

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A367676 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 the version of the Eden growth model described in A367671 when n square cells have been added.

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A367764 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 in the Eden growth model on the square lattice (see A367760), when n square cells have been added.

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A367765 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 the Eden growth model on the square lattice (see A367760), when n square cells have been added.

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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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A368001 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 as the image of a simple random walk on the square lattice.

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A368392 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 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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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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