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 1-5 of 5 results.

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).

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

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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).

A255890 Polyomino Family Planners: a(n) is the least number of children of a polyomino of size n.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 3, 3, 3, 2, 3, 4, 2, 2, 4, 4, 2, 3, 4, 4, 3, 3, 5, 4, 2, 3, 5, 5, 3, 3, 5, 6, 3, 3, 5, 6, 3
Offset: 0

Views

Author

Gordon Hamilton, Mar 09 2015

Keywords

Comments

For n = (2k+1)^2 + (2k)^2, a(n) = k+1 and a(m) > k+1 for m > n.
This is a beautiful exploration of symmetry for the elementary classroom.
A "child" is any polyomino formed by adjoining a cell at any edge. - N. J. A. Sloane, Mar 10 2015
Optimal polyominoes have at least fourfold symmetry. - Charlie Neder, Mar 03 2019

Examples

			a(7) = 3 because this polyomino has only three children:
   xx                        xxx      xx       xx
   xxx      has children     xxx      xxxx     xxx
    xx                        xx       xx       xxx
a(8) = 3 because of this polyomino:
                             xxxx
                             xxxx
a(9) = 2 because of this polyomino:
                             xxx
                             xxx
                             xxx
a(10) = 3 because of this polyomino (not the 2*5 rectangle):
                             xx
                             xxx
                              xxx
                               xx
a(11) = 4 because of this polyomino:
                             xxx
                            xxxxx
                             xxx
a(12) = 2 because of this polyomino:
                             xx
                            xxxx
                            xxxx
                             xx
a(13) = 2 because of the following polyomino. This will be the last time 2 will be encountered in the sequence (see comments above):
                              x
                             xxx
                            xxxxx
                             xxx
                              x
a(14) = 4 because of this polyomino:
                             xxx
                            xxxx
                            xxxx
                            xxx
a(15) = 4 because of this polyomino:
                            xx
                            xxxx
                             xxx
                             xxxx
                               xx

Crossrefs

Cf. A000105, A255894, A367758.
Row minima of A367443 (for n>=1).

Formula

From Charlie Neder, Mar 03 2019: (Start)
a(4k) >= b, where b is the least integer such that b(2b-1) >= k.
a(4k+1) = c, where c is the least integer such that (c-1)(2c-1) >= k. (End)

Extensions

a(16)-a(36) from Charlie Neder, Mar 03 2019

A367439 a(n) is the degree of the polyomino with binary code A246521(n+1) in the polyomino graph PG(n) defined in A367435.

Original entry on oeis.org

0, 0, 1, 1, 4, 3, 4, 3, 2, 10, 8, 3, 9, 10, 9, 8, 9, 10, 8, 4, 2, 15, 28, 15, 12, 12, 10, 17, 14, 19, 20, 15, 14, 15, 13, 18, 20, 9, 14, 13, 17, 4, 12, 16, 18, 11, 9, 10, 15, 22, 19, 10, 19, 14, 16, 3, 36, 36, 35, 31, 28, 30, 36, 22, 29, 37, 16, 11, 28, 13, 24
Offset: 1

Views

Author

Pontus von Brömssen, Nov 18 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), where the intermediate (the set of cells remaining when the cell to be moved is detached) is required to be a (connected) polyomino.
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, 8, 3, 9, 10, 9, 8, 9, 10, 8, 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, A246521, A367126, A367435, A367437 (row maxima), A367438, A367443.

Formula

a(n) <= A367126(n).

A367441 Minimum size of a set of polyominoes with n-1 cells such that all free polyominoes with n cells can be obtained by adding one cell to one of the polyominoes in the set.

Original entry on oeis.org

1, 1, 2, 4, 8, 19
Offset: 2

Views

Author

Pontus von Brömssen, Nov 18 2023

Keywords

Comments

a(8) <= 54, a(9) <= 160.
Apparently, a(n) is close to A365621(n+1) for n <= 8. Is this just a coincidence?

Examples

			For n <= 4, all polyominoes with n-1 cells are needed to obtain all polyominoes with n cells by adding one cell, so a(n) = A000105(n-1).
For n = 5, all but the square tetromino are needed to obtain all pentominoes, so a(5) = A000105(4)-1 = 4.
For n = 6, there are 5 different sets of a(6) = 8 pentominoes that are sufficient to obtain all hexominoes. One of these sets consists of the I, L, N, P, U, V, W, and Y pentominoes. The X pentomino is the only pentomino that does not appear in any of these sets. The I, L, N, and W pentominoes are needed in all such sets.
For n = 7, there are 8 different sets of a(7) = 19 hexominoes that are sufficient to obtain all heptominoes. 14 hexominoes appear in all these sets, 10 appear in none of them.

Links

Crossrefs

Cf. A000105, A365621, A367435, A367440, A367443.
Showing 1-5 of 5 results.

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