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 1-6 of 6 results.

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).

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.

A367762 Numerator of the greatest probability that a particular free polyomino with n cells appears in the Eden growth model (see A367760).

Original entry on oeis.org

1, 1, 2, 1, 2, 107, 70037, 813359, 1523168309, 1248684827, 26754412658849, 21916760758464961, 967387787846413936477, 271537923398149191011927
Offset: 1

Views

Author

Pontus von Brömssen, Dec 02 2023

Keywords

Comments

a(n) is the numerator of the maximum of A367760/A367761 over the n-th row.

Examples

			For 1 <= n <= 14, the following are all polyominoes that have the maximum probabilities for their respective sizes. Except for n = 7, there is just one such polyomino.
                    _      _      _
        _    _     | |    | |_   | |_ _
   _   | |  | |_   | |_   |   |  |    _|
  |_|  |_|  |_ _|  |_ _|  |_ _|  |_ _|
            _                 _      _
   _ _     | |_    _ _      _| |_   | |_ _
  |   |    |   |  |   |_   |     |  |     |
  |   |_   |   |  |     |  |    _|  |     |
  |_ _ _|  |_ _|  |_ _ _|  |_ _|    |_ _ _|
   _ _        _ _      _ _ _      _ _ _
  |   |_    _|   |    |     |    |     |
  |     |  |     |_   |     |_   |     |_
  |     |  |      _|  |      _|  |       |
  |_ _ _|  |_ _ _|    |_ _ _|    |_ _ _ _|

Links

Crossrefs

Cf. A367673, A367760, A367761, A367763 (denominators), A367766.

A367767 Denominator of the greatest probability that a particular fixed polyomino with n cells appears in the Eden growth model (see A367760).

Original entry on oeis.org

1, 2, 6, 6, 20, 1200, 3024000, 63504000, 5334336000, 4779565056000, 9635603152896000, 404695332421632000, 44071321700715724800000, 7329942225263039348736000000
Offset: 1

Views

Author

Pontus von Brömssen, Dec 02 2023

Keywords

Comments

a(n) is the denominator of the maximum of A367764/A367765 over the n-th row.

Links

Crossrefs

Cf. A367678, A367760, A367763, A367764, A367765, A367766 (numerators).

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).

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

Views

Author

Pontus von Brömssen, Dec 22 2023

Keywords

Comments

a(n) is the numerator of the maximum of A368392/A368393 over the n-th row.
See A368386 for details.

Links

Crossrefs

Cf. A367677, A367766, A368004, A368386, A368390, A368392, A368393, A368395 (denominators), A368865 (external diffusion-limited aggregation).
Showing 1-6 of 6 results.

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