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 1-3 of 3 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).

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).

A367440 Independence number of the polyomino graph PG(n) defined in A367435.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 18, 46
Offset: 1

Views

Author

Pontus von Brömssen, Nov 18 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, where the intermediate (the set of cells remaining when the cell to be moved is detached) is required to be a (connected) polyomino.

Links

Crossrefs

Cf. A367127, A367435, A367436.

Formula

a(n) >= A367127(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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