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.

A006746 Number of axially symmetric polyominoes with n cells.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 9, 23, 38, 90, 147, 341, 564, 1294, 2148, 4896, 8195, 18612, 31349, 70983, 120357, 271921, 463712, 1045559, 1792582, 4034832, 6950579, 15619507, 27023509, 60638559, 105320716, 236006955, 411364068, 920626423, 1609836928
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid. - Joshua Zucker, Mar 08 2008
The line of reflective symmetry may pass through the center of a square or a vertex of a square. These subsets are enumerated by A349328 and A349329 respectively. - John Mason, Feb 17 2022

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000105, A001168, A030227, A361625.
Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A349328, A349329.

Formula

a(n) = A349328(n) + A349329(n/2) for even n, otherwise a(n) = A349328(n). - John Mason, Feb 17 2022

Extensions

Extended to n=28 by Tomás Oliveira e Silva

A346799 Number of fixed polyominoes with n cells that have a horizontal axis of symmetry that passes through the centers of cells.

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 24, 36, 86, 133, 314, 499, 1164, 1888, 4366, 7192, 16522, 27548, 62954, 106004, 241203, 409492, 928376, 1587151, 3586999, 6169400, 13904736, 24041597, 54053950, 93896826, 210654990, 367450477, 822754494
Offset: 1

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Author

Robert A. Russell, Aug 04 2021

Keywords

Comments

This is one of three sequences needed to calculate the number of achiral polyominoes, A030227. The three sequences together contain exactly two copies of each achiral polyomino. This is the FL sequence in the Shirakawa link. The sequence can be quickly calculated using Redelmeier's method; each polyomino cell in the lowest row is counted as one, while all the other polyomino cells are counted as two. Jensen's transfer matrix method (see Knuth POLYNUM program) could be modified to enumerate this sequence for over 100 terms; one needs only to keep track of the number of polyomino cells in the original row.
John Mason has pointed out that a(n) is also the number of achiral (2n)-ominoes with twofold rotational symmetry centered at the center of an edge. Just add to each polyomino its reflection in its leftmost edge to obtain these, the subset of A056877 with edge centers. - Robert A. Russell, Dec 15 2021

Examples

			For a(5)=7, the polyominoes are:    X
X       X   XX   XX    X            X
XXX   XXX   X     X   XXX   XXXXX   X
X       X   XX   XX    X            X
                                    X

Links

Crossrefs

Cf. A030227, A056877, A346800.

Formula

a(n) = A351127(n) + 2 * A351190(n) + A346799(n / 2) + 2 * A349328(n), setting A346799(n / 2) = 0 for noninteger arguments. - John Mason, Mar 13 2023

A349329 Number of polyominoes with 2n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the corner of at least one square.

Original entry on oeis.org

0, 0, 2, 7, 28, 100, 368, 1335, 4912, 18125, 67477, 252573, 951363, 3601113, 13695150, 52291510, 200389661, 770391542, 2970337861, 11482318605, 44491635790, 172766013959, 672185703574, 2619993338628
Offset: 1

Views

Author

John Mason, Nov 15 2021

Keywords

Examples

			a(3)=2 because of hexominoes:
  OO   and   O
  O          OO
  O          OO
  OO         O

Links

Crossrefs

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A349328.

Formula

a(n) = A006746(2n) - A349328(2n).
a(n) = 2*(A006747(n) + A006748(n)) + A144553(n) + A056878(n) + A006746(n) + 4*A006749(n). - corrected by John Mason, Feb 26 2023

Extensions

Name corrected by John Mason, Feb 01 2022

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.
Showing 1-4 of 4 results.

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

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