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.

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A366009 Number of free polyominoids with n cells, allowing any corner-connections and right-angled edge-connections.

Original entry on oeis.org

1, 3, 30, 531, 13095
Offset: 1

Views

Author

Pontus von Brömssen, Sep 28 2023

Keywords

Comments

See A365995 for further details and a table of related sequences.

Links

Crossrefs

Cf. A365995, A366010 (fixed).
29th row of A366766.

A366336 Number of free (4,3)-polyominoids with n cells.

Original entry on oeis.org

1, 2, 9, 67, 714, 10974
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

See A366334 for definitions.

Links

Crossrefs

78th row of A366766.
Cf. A366334, A366337 (fixed).

A361625 Number of free polyominoes with checkerboard-pattern-colored vertices with n cells.

Original entry on oeis.org

1, 1, 3, 7, 20, 60, 204, 702, 2526, 9180, 33989, 126713, 476597, 1802109, 6850969, 26151529, 100207548, 385217382, 1485216987, 5741240989, 22246000726, 86383317470, 336093551268, 1309997856337, 5114452295933, 19998171631076, 78306014924606, 307022177714062
Offset: 1

Views

Author

Andrey Zabolotskiy, Mar 19 2023; thanks to John Mason for his help

Keywords

Comments

Also, number of polysticks of size n (see A019988), with the requirement that any two sticks are connected by a sequence of adjacent, alternately horizontal and vertical sticks. - Pontus von Brömssen, Sep 01 2023

Examples

			There are 2 ways to color the 4 corners of a monomino with black and white colors alternatingly, but they are related by a rotation or a reflection, so a(1) = 1. a(2) is also 1 because the two ways to color the 6 vertices of a domino with black and white colors in the checkerboard pattern are related to each other by a reflection or a rotation. The same is true for the stick tromino, but the two ways to color the 8 vertices of the L-tromino are inequivalent, so a(3) = 3.
For n = 3, the a(3) = 3 allowed polysticks are:
  _     _
  _|  _|   _|_

Links

Crossrefs

A122675 is the 3-dimensional analog based on polycubes.
Cf. A001933, A019988, A359689.
Cf. A000105, A351190, A351142, A351127, A349328, A346799, A234008.
5th row of A366766.

Formula

a(n) = 2 * A000105(n) - (A351190(n) + A351142(n) + A351127(n) + A349328(n) + A346799(n/2) + A234008(n/2)), where the last two terms are only included if 2|n. I.e., every free polyomino is counted twice here unless it is symmetric with respect to a Pi/2 rotation centered at a cell, or a Pi rotation centered at an edge, or a reflection with respect to an axis parallel to the grid and passing through cells.

A366338 Number of free (4,2)-polyominoids with n cells, allowing right-angled (or hard) connections only.

Original entry on oeis.org

1, 1, 8, 44, 509, 7091
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

Two squares sharing an edge have a right-angled connection if they do not lie in the same plane.
Connections that are not right-angled (flat connections) may occur, but the polyominoids considered here must be connected through right-angled connections only.

Links

Crossrefs

45th row of A366766.
Cf. A075679, A365654, A366334, A366339 (fixed), A366340.

A366340 Number of free (4,3)-polyominoids with n cells, allowing right-angled (or hard) connections only.

Original entry on oeis.org

1, 1, 5, 19, 123, 954, 9324
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

Two cubes sharing a face have a right-angled connection if they do not lie in the same 3-dimensional affine subspace.
Connections that are not right-angled (flat connections) may occur, but the polyominoids considered here must be connected through right-angled connections only.
Also, number of free polysticks in 4 dimensions with right-angled connections.

Links

Crossrefs

41st and 77th row of A366766.
Cf. A075679, A365654, A366336, A366338, A366341 (fixed).

A365353 Number of free corner-connected 4-dimensional polyhypercubes with n cells.

Original entry on oeis.org

1, 1, 4, 23, 207, 2794
Offset: 1

Views

Author

Pontus von Brömssen, Sep 02 2023

Keywords

Links

Crossrefs

147th row of A366766.
Cf. A038171, A068870.
See A365366 for a table of similar sequences.

A365354 Number of free edge-connected 4-dimensional polyhypercubes with n cells.

Original entry on oeis.org

1, 1, 6, 84, 2363
Offset: 1

Views

Author

Pontus von Brömssen, Sep 02 2023

Keywords

Links

Crossrefs

143rd row of A366766.
Cf. A038173, A068870.
See A365366 for a table of similar sequences.

A365355 Number of free 4-dimensional polyhypercubes with n cells, allowing corner- and edge-connections.

Original entry on oeis.org

1, 2, 14, 305, 12819
Offset: 1

Views

Author

Pontus von Brömssen, Sep 02 2023

Keywords

Links

Crossrefs

151st row of A366766.
Cf. A068870, A363206.
See A365366 for a table of similar sequences.

A365356 Number of free face-connected 4-dimensional polyhypercubes with n cells.

Original entry on oeis.org

1, 1, 5, 36, 407, 7579
Offset: 1

Views

Author

Pontus von Brömssen, Sep 03 2023

Keywords

Links

Crossrefs

141st row of A366766.
Cf. A007900 (coordination sequence of the underlying graph), A038119, A068870.
See A365366 for a table of similar sequences.

A365357 Number of free 4-dimensional polyhypercubes with n cells, allowing corner- and face-connections.

Original entry on oeis.org

1, 2, 11, 139, 3196
Offset: 1

Views

Author

Pontus von Brömssen, Sep 03 2023

Keywords

Links

Crossrefs

149th row of A366766.
Cf. A068870, A117216 (coordination sequence of the underlying graph), A363205.
See A365366 for a table of similar sequences.
Previous Showing 31-40 of 48 results. Next

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

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