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.

A367124 Maximum degree of the n-omino graph defined in A098891.

Original entry on oeis.org

0, 0, 1, 4, 10, 28, 39, 68, 81, 116, 140, 186, 204
Offset: 1

Views

Author

Pontus von Brömssen, Nov 05 2023

Keywords

Comments

Largest number of free polyominoes that can be made from a polyomino with n cells by moving one of its cells (not counting itself).

Examples

			For 1 <= n <= 12, the following polyominoes have the maximum degree in the polyomino graph of their respective sizes (see also link):
                                                _             _
             _           _        _    _      _| |     _     | |
        _   | |   _     | |     _| |  | |_   |_  |   _| |    | |_
   _   | |  | |  | |_   | |_   |_  |  |   |    | |  |_  |_   |   |
  |_|  |_|  |_|  |_ _|  |_ _|    |_|  |_ _|    |_|    |_ _|  |_ _|
                     _          _      _        _
     _      _       | |       _| |    | |      | |_      _ _
   _| |    | |      | |      |_  |_   | |_ _   |   |_   |   |_ _
  |_  |_   | |_ _   | |_ _     |   |  |  _  |  |  _  |  |  _ _  |
    |   |  |     |  |     |    |   |  | |_| |  | |_| |  | |_ _| |
    |_ _|  |_ _ _|  |_ _ _|    |_ _|  |_ _ _|  |_ _ _|  |_ _ _ _|

Links

Crossrefs

Cf. A098891, A367123, A367125, A367437.
Row maxima of A367126.

Formula

a(n) >= A367437(n).

A367126 a(n) is the degree of the polyomino with binary code A246521(n+1) in the n-omino graph defined in A098891.

Original entry on oeis.org

0, 0, 1, 1, 4, 3, 4, 3, 2, 10, 9, 5, 9, 10, 9, 8, 9, 10, 9, 4, 2, 16, 28, 16, 14, 12, 12, 18, 15, 20, 21, 16, 16, 16, 15, 18, 20, 11, 14, 13, 18, 6, 12, 16, 18, 11, 9, 11, 15, 22, 20, 11, 19, 14, 16, 3, 38, 36, 35, 33, 31, 32, 38, 25, 31, 38, 17, 14, 30, 14, 26
Offset: 1

Views

Author

Pontus von Brömssen, Nov 05 2023

Keywords

Comments

Number of free polyominoes that can be made from the polyomino with binary code A246521(n+1) by moving one of its cells (not counting itself).
Can be read as an irregular triangle, whose m-th row contains A000105(m) terms, m >= 1.

Examples

			As an irregular triangle:
   0;
   0;
   1, 1;
   4, 3, 4, 3,  2;
  10, 9, 5, 9, 10, 9, 8, 9, 10, 9, 4, 2;
  ...
For n = 8, A246521(8+1) = 30 is the binary code of the S-tetromino. By moving one cell of the S-tetromino, we can obtain the L, O, and T tetrominoes (but not the I tetromino), so a(8) = 3.

Links

Crossrefs

Cf. A000105, A098891, A246521, A367123, A367124 (row maxima), A367125, A367439, A367443.

Formula

a(n) >= A367439(n).

A367125 Number of n-ominoes that have the maximum degree (A367124(n)) in the n-omino graph defined in A098891.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 1, 1, 2, 1, 1, 1, 10
Offset: 1

Views

Author

Pontus von Brömssen, Nov 05 2023

Keywords

Links

Crossrefs

Cf. A098891, A367123, A367124, A367438.

A367127 Independence number of the n-omino graph defined in A098891.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 17
Offset: 1

Views

Author

Pontus von Brömssen, Nov 05 2023

Keywords

Comments

Maximum size of a set of free polyominoes with n cells such that no polyomino in the set can be obtained from another by moving one of its cells.
a(8) is either 45 or 46.

Links

Crossrefs

Cf. A098891, A367123, A367124, A367440.

Formula

a(n) <= A367440(n).

A367436 Number of Hamiltonian cycles in the polyomino graph PG(n) defined in A367435.

Original entry on oeis.org

1, 1, 0, 2, 4080
Offset: 1

Views

Author

Pontus von Brömssen, Nov 18 2023

Keywords

Comments

A cycle and its reverse are not both counted.
We follow the convention in A003216 that the complete graphs on 1 and 2 nodes have 1 and 0 Hamiltonian cycles, respectively, so that a(1) = a(2) = 1 and a(3) = 0, but it could also be argued that a(1) = a(2) = 0 and/or a(3) = 1.

Links

Crossrefs

Cf. A003216, A367123, A367435.

Formula

a(n) <= A367123(n).
a(n) > 0 for 4 <= n <= 13.

A365621 Minimum size of a set of polyominoes with n cells such that all other free polyominoes with n cells can be obtained by moving one cell of one of the polyominoes in the set.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7
Offset: 1

Views

Author

Pontus von Brömssen, Nov 14 2023

Keywords

Comments

a(n) is the domination number of the n-omino graph defined in A098891.
The intermediate (the set of cells remaining when the cell to be moved is detached) does not have to be a connected (n-1)-omino.
a(8) <= 18, a(9) <= 53.
Apparently, a(n) is close to A367441(n-1) for 3 <= n <= 9. Is this just a coincidence?

Examples

			For n <= 3, any one polyomino with n cells is enough to construct the others (if any) by moving one cell, so a(n) = 1.
For n = 4, either the L or the T tetromino suffices to construct the other four, so a(4) = 1.
Below are examples of sets of a(n) polyominoes that are sufficient to construct all other polyominoes with n cells, 5 <= n <= 7:
   _
  | |     _
  | |_   | |_
  |  _|  |   |
  |_|    |_ _|
   _
  | |     _
  | |    | |       _ _
  | |    | |_    _|  _|
  | |_   |   |  |   |
  |_ _|  |_ _|  |_ _|
   _
  | |                                  _
  | |     _        _          _       | |           _        _
  | |    | |_     | |_       | |      | |_      _ _| |      | |
  | |_   |  _|    |_  |     _| |_    _|  _|   _|  _ _|   _ _| |_
  |  _|  | |_ _    _| |_   |    _|  |  _|    |  _|      |  _ _ _|
  |_|    |_ _ _|  |_ _ _|  |_ _|    |_|      |_|        |_|

Links

Crossrefs

Cf. A098891, A367123, A367124, A367127, A367441.
Showing 1-6 of 6 results.

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