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 1-4 of 4 results.

A385381 Triangle read by rows: T(n,k) is the number of polyominoes, i.e., connected nonempty subsets of square cells (or vertices), of the n X k flat torus, up to cyclic shifts and reflections of rows and columns; 1 <= k <= n.

Original entry on oeis.org

1, 2, 5, 3, 10, 28, 4, 21, 102, 801, 5, 40, 382, 6790, 129550, 6, 86, 1788, 68569, 2694721
Offset: 1

Views

Author

Pontus von Brömssen, Jun 27 2025

Keywords

Comments

Two square cells are connected if they share an edge.
Subsets that differ by interchanging rows and columns (when n = k) are considered distinct. For example, there are two polyominoes of size 2 when 2 <= k <= n, one horizontal and one vertical. See A385384 for the analog of the main diagonal of this sequence in the case where such subsets are considered identical.

Examples

			Triangle begins:
  n\k| 1  2    3     4       5  6
  ---+---------------------------
  1  | 1
  2  | 2  5
  3  | 3 10   28
  4  | 4 21  102   801
  5  | 5 40  382  6790  129550
  6  | 6 86 1788 68569 2694721  ?
		

Crossrefs

Cf. A385382 (main diagonal), A385384, A385386 (edge subsets).

A385382 Number of polyominoes, i.e., connected nonempty subsets of square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns.

Original entry on oeis.org

1, 5, 28, 801, 129550
Offset: 1

Views

Author

Pontus von Brömssen, Jun 27 2025

Keywords

Crossrefs

Main diagonal of A385381.
Row sums of A385383.
Cf. A385384 (interchange of rows and columns of the torus allowed), A385387 (edge subsets).

A385389 Number of polysticks, i.e., connected nonempty subsets of edges, of the n X n flat torus, up to cyclic shifts and reflections of rows and columns, as well as interchange of rows and columns.

Original entry on oeis.org

2, 23, 3257, 21961750
Offset: 1

Views

Author

Pontus von Brömssen, Jun 27 2025

Keywords

Crossrefs

Row sums of A385390.
Cf. A385384 (polyominoes), A385387 (interchange of rows and columns of the torus not allowed).

A385385 Irregular triangle read by rows: T(n,k) is the number of polyominoes of size k, i.e., connected subsets of k square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns, as well as interchange of rows and columns; 1 <= k <= n^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 2, 1, 1, 1, 1, 2, 5, 10, 23, 44, 80, 87, 86, 49, 32, 10, 5, 1, 1, 1, 1, 2, 5, 12, 32, 88, 249, 675, 1699, 3747, 6993, 10538, 12531, 11580, 8458, 4975, 2378, 943, 305, 87, 19, 5, 1, 1
Offset: 1

Views

Author

Pontus von Brömssen, Jun 27 2025

Keywords

Comments

For n = 4, there are 384 automorphisms of the 4 X 4 torus grid graph (it is isomorphic to the 4-dimensional hypercube graph), but here we only consider the subgroup consisting of the 128 symmetries of the 4 X 4 torus. Using the full automorphism group of the torus grid graph would change row 4 to the corresponding row of A369605.

Examples

			Triangle begins:
  1;
  1, 1, 1, 1;
  1, 1, 2, 3,  4,  4,  2,  1,  1;
  1, 1, 2, 5, 10, 23, 44, 80, 87, 86, 49, 32, 10, 5, 1, 1;
  ...
		

Crossrefs

Cf. A000105, A369605, A385383 (interchange of rows and columns of the torus not allowed), A385384 (row sums), A385390 (edge subsets).

Formula

T(n,k) = A000105(k) if n >= k.
T(n,k) >= A385383(n,k)/2, with equality if and only if k = 2.
Showing 1-4 of 4 results.