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 1-7 of 7 results.

A367123 Number of Hamiltonian cycles in the n-omino graph defined in A098891.

Original entry on oeis.org

1, 1, 0, 2, 16800
Offset: 1

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Author

Pontus von Brömssen, Nov 05 2023

Keywords

Comments

The n-omino graph has all A000105(n) free n-ominoes as nodes, and two n-ominoes are joined by an edge if one can be obtained from the other by moving one cell. The intermediate is allowed not to be a connected (n-1)-omino; for example, there is an edge between the V and W pentominoes, but to transform one to the other the central cell must be moved, and the remaining 4 cells is not a tetromino.
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.

Examples

			For n = 4, there are a(4) = 2 Hamiltonian cycles in the tetromino graph: I-L-O-S-T-I and I-L-S-O-T-I, using conventional names of the tetrominoes.
For n = 5, one of the a(5) = 16800 Hamiltonian cycles in the pentomino graph is I-L-P-U-V-T-N-W-Z-F-X-Y-I.
See links for an example for n = 6.

Links

Crossrefs

Cf. A000105, A003216, A098891, A367124, A367125, A367126, A367127, A367436.

Formula

a(n) > 0 for 4 <= n <= 13.
a(n) >= A367436(n).

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).

A367435 Let PG(n) be the graph with one node for each free n-celled polyomino and edges between nodes corresponding to polyominoes that can be obtained from each other by moving one cell, where the intermediate (the set of cells remaining when the cell to be moved is detached) is required to be a (connected) polyomino. a(n) is the number of edges in PG(n).

Original entry on oeis.org

0, 0, 1, 8, 45, 254, 1258, 6181, 28062, 125714, 550402, 2394654, 10326665
Offset: 1

Views

Author

Pontus von Brömssen, Nov 18 2023

Keywords

Comments

Equivalently, there is an edge between two nodes if the corresponding n-celled polyominoes can be obtained from the same (n-1)-celled polyomino by adding one cell.
In the n-omino graph defined in A098891, the intermediate is not required to be a polyomino, so PG(n) is a spanning subgraph of that graph. For n = 5, for example, there is an edge between the V and W pentominoes in the graph in A098891, but not in PG(5).

Links

Crossrefs

Half the row sums of A367439.
Cf. A000105, A098891, A367436, A367437, A367438, A367440, A367441, A367443.

Formula

a(n) <= A098891(n).

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 1-7 of 7 results.

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