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.

A367671 a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in a version of the Eden growth model on the square lattice, when n square cells have been added.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 23, 4, 1, 253, 5, 1, 23, 713, 11, 5, 149, 157, 5, 23, 1, 3671, 286417, 16, 73, 289, 1, 2657, 103, 289, 15923, 19067, 1, 1661, 1, 10019, 16591, 1, 323, 193, 1661, 2, 169, 14603, 71, 853, 11, 23, 1037, 27151, 15923, 23, 529, 487, 14267, 1
Offset: 1

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Author

Pontus von Brömssen, Nov 26 2023

Keywords

Comments

In the Eden growth model, there is a single initial unit square cell in the plane and more squares are added one at a time, selected randomly among those squares that share an edge with one of the already existing squares. In the version considered here, all such new squares have the same probability of being selected, whereas in Eden (1961) it appears that the probability is proportional to the number of already existing squares with which the new square shares an edge. See A367760 for the latter version.
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;
    5, 2, 23,  4,   1;
  253, 5,  1, 23, 713, 11, 5, 149, 157, 5, 23, 1;
  ...
For n = 7, the T-tetromino has binary code A246521(n+1) = 27. It can be obtained either via the straight tromino (probability 1/3 * 1/4) or via the L-tromino (probability 2/3 * 2/7), so the probability of obtaining the T-tetromino is 1/12 + 4/21 = 23/84 and a(7) = 23.

Links

Crossrefs

Cf. A000105, A246521, A335573, A367672 (denominators), A367673, A367675, A367676, A367760.

Formula

a(n)/A367672(n) = (A367675(n)/A367676(n))*A335573(n+1).

A367672 a(n) is the denominator of the probability that 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, 1, 3, 3, 14, 21, 84, 21, 12, 1008, 126, 21, 315, 5040, 126, 126, 2016, 1008, 126, 672, 60, 99792, 4989600, 1155, 3780, 9072, 66, 30240, 3360, 4536, 554400, 453600, 60, 45360, 60, 277200, 498960, 66, 5184, 9072, 45360, 189, 13860, 554400, 4620, 50400, 1260, 3465, 73920, 712800, 554400, 3465, 12960, 12600, 453600, 360
Offset: 1

Views

Author

Pontus von Brömssen, Nov 26 2023

Keywords

Comments

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.

Examples

			As an irregular triangle:
     1;
     1;
     3,   3;
    14,  21, 84,  21,   12;
  1008, 126, 21, 315, 5040, 126, 126, 2016, 1008, 126, 672, 60;
  ...

Links

Crossrefs

Cf. A000105, A246521, A335573, A367671 (numerators), A367674, A367675, A367676, A367761.

Formula

A367671(n)/a(n) = (A367675(n)/A367676(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

Views

Author

Pontus von Brömssen, Nov 26 2023

Keywords

Comments

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.

Examples

			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;
  ...

Links

Crossrefs

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

Formula

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

Views

Author

Pontus von Brömssen, Dec 02 2023

Keywords

Comments

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.

Examples

			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;
  ...

References

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

Links

Crossrefs

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

Formula

a(n)/A367765(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

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).

A367677 Numerator of the greatest probability that a particular fixed polyomino with n cells appears in the version of the Eden growth model described in A367671.

Original entry on oeis.org

1, 1, 1, 2, 23, 289, 254179, 777607, 22699340513, 480319839870583, 2390156990671551019, 263173875833094285221, 7370729029770126053601007351, 20600083403482483161475107845607517
Offset: 1

Views

Author

Pontus von Brömssen, Nov 26 2023

Keywords

Comments

a(n) is the numerator of the maximum of A367675/A367676 over the n-th row.

Examples

			For 1 <= n <= 14, 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. A367671, A367673, A367675, A367676, A367678 (denominators), A367766.

A367678 Denominator of the greatest probability that a particular fixed polyomino with n cells appears in the version of the Eden growth model described in A367671.

Original entry on oeis.org

1, 2, 6, 21, 672, 18144, 32659200, 204120000, 12243879648000, 550274933894400000, 5577105239386480800000, 870028417344291004800000, 51097100257451535417918627840000, 292528465778659685432564243496960000000
Offset: 1

Views

Author

Pontus von Brömssen, Nov 26 2023

Keywords

Comments

a(n) is the denominator of the maximum of A367675/A367676 over the n-th row.

Links

Crossrefs

Cf. A367671, A367674, A367675, A367676, A367677 (numerators), A367767.

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

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

Links

Crossrefs

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

Formula

a(n)/A368393(n) = (A368386(n)/A368387(n))/A335573(n+1).
Showing 1-8 of 8 results.

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