A349329 -id:A349329 - cp's OEIS frontend

A006746 Number of axially symmetric polyominoes with n cells.

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

N. J. A. Sloane

nonn

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

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

John Mason, Table of n, a(n) for n = 1..50
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...

Cf. A000105, A001168, A030227, A361625.

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

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

Extended to n=28 by Tomás Oliveira e Silva

A234006 Free polyominoes with 2n squares, having reflectional symmetry on axis that coincides with edges.

Original entry on oeis.org

1, 2, 4, 11, 35, 114, 392, 1381, 4998, 18292, 67791, 253182, 952527, 3603389, 13699516, 52300071, 200406183, 770424072, 2970400815, 11482442855, 44491876993, 172766491178, 672186631950, 2619995178793, 10228902801505, 39996341268584, 156612023001490, 614044347934591

Offset: 1

John Mason, Dec 18 2013

nonn

The number of free polyominoes of size 2n that have reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.

John Mason, Table of n, a(n) for n = 1..50

Cf. A000105, A001168, A001933, A151525, A182645, A234007, A234008, A234009, A234010, A349329, A346799, A346800, A351191.

A151525 = Cases[Import["https://oeis.org/A151525/b151525.txt", "Table"], {, }][[All, 2]];
A182645 = Cases[Import["https://oeis.org/A182645/b182645.txt", "Table"], {, }][[All, 2]];
A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {, }][[All, 2]];
a[n_] := If[OddQ[n], A151525[[n]], A151525[[n]] + A182645[[n/2]] - A001168[[n/2]]];
Array[a, 28] (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)

a(2*n+1) = A151525(2*n+1), a(2*n) = A151525(2*n) + A182645(n) - A001168(n). - Andrew Howroyd, Dec 05 2018

If n odd, a(n) = A349329(n) + A346799(n), otherwise a(n) = A349329(n) + A346799(n) + A346800(n/2) + A351191(n/2). - John Mason, Mar 15 2023

a(12)-a(28) from Andrew Howroyd, Dec 05 2018

A349328 Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the center of at least one square.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 9, 16, 38, 62, 147, 241, 564, 926, 2148, 3561, 8195, 13700, 31349, 52858, 120357, 204444, 463712, 792986, 1792582, 3083469, 6950579, 12018394, 27023509, 46943409, 105320716, 183715445, 411364068, 720236762, 1609836928, 2828102115

Offset: 1

John Mason, Nov 15 2021

nonn

			a(4) is 1 because of the tetromino:
   O
  OOO

John Mason, Table of n, a(n) for n = 1..50

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

For odd n, a(n) = A006746(n).

For even n, a(n) = A006746(n) - A349329(n/2).

A331621 Number of distinct structures that can be made from n cubes without overhangs.

Original entry on oeis.org

1, 1, 2, 4, 12, 35, 129, 495, 2101, 9154, 41356, 189466, 880156, 4120515, 19425037, 92038062, 438030079, 2092403558, 10027947217, 48198234188, 232261124908, 1121853426115, 5430222591596

Offset: 0

Nicholas A. Kennedy, Jan 22 2020

This is an extension of the free polyominoes (A000105) to the third dimension. Structures are considered equivalent if they can be mapped into each other by reflection in a vertical plane or rotation around the vertical axis. They are not equivalent if they can only be mapped onto each other by rotation around an axis parallel to the horizontal plane.
From John Mason, Mar 03 2025: (Start)
Equivalently, the sequence enumerates inscribed polyominoes that have a positive integer in each square, such that the size of the polyomino is considered to be the sum of the integers.
Examples.
  Size 1:
  +-+
  |1|
  +-+
.
  Size 2:
  +-+-+ +-+
  |1|1| |2|
  +-+-+ +-+
.
  Size 3:
  +-+-+-+ +-+-+ +-+-+ +-+
  |1|1|1| |2|1| |1|1| |3|
  +-+-+-+ +-+-+ +-+-+ +-+
                  |1|
                  +-+
(End)

			For n = 0, one (the empty) structure is possible.
For n = 1, only one structure is possible, a single cube.
For n = 2, two structures are possible: two cubes one on top of the other, and two next to each other.
For n = 3, four structures are possible: an L shape with the L oriented vertically, an L shape with the L laid flat, a structure with 3 cubes stacked on top of each other and a structure with 3 cubes laid flat.
For n = 4, there are 12 possible distinct structures made from 4 cubes without overhangs. These include 1 structure that is 4 cubes tall, 1 structure that is 3 cubes tall, 5 structures that are 2 cubes tall and 5 that are 1 cube tall.

Arnauld Chevallier, Javascript Node.JS program
Nicholas A. Kennedy, Buildings made from cubes, Code Golf Stack Exchange 2020
Nicholas A. Kennedy, Images of possible structures made from 4 cubes
Miles, Java program

Extension to the third dimension of A000105.

Cf. A006749, A007318, A259090, A349329, A351616, A234008.

Java
```
See Miles link
```
JavaScript
```
See Arnauld Chevallier link
```

From John Mason, Mar 03 2025: (Start)
Define a(n)=f(n)+g(n) where f(n) enumerates the structures having an asymmetrical base, and g(n) enumerates the structures having a symmetrical base.
Then for n>=4, f(n) = Sum_{i=4..n} ((A006749(i)*C(n-1,i-1)), and g(n) < Sum_{i=1..n} ((A259090(i)*C(n-1,i-1)).
For structures having a base with: reflective orthogonal symmetry about an axis that passes through cell vertices, 180 degree rotational symmetry about a point at a cell vertex or midway along an edge, the number of structures is (1) for odd n: Sum_{i=1..n} ((S(i)*C(n-1,i-1)) / 2, and (2) for even n: Sum_{i=1..n} ((S(i)*(C(n-1,i-1)+C(n/2-1,i/2-1)) / 2, for S(n) = A349329(n), A351616(n) and A234008(n) respectively.
As almost all polyominoes are asymmetrical, a(n)/f(n) tends to 1. (End)

a(15)-a(22) from John Mason, Mar 03 2025

cp's OEIS Frontend

A006746 Number of axially symmetric polyominoes with n cells.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A234006 Free polyominoes with 2n squares, having reflectional symmetry on axis that coincides with edges.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A349328 Number of polyominoes with n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the center of at least one square.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A331621 Number of distinct structures that can be made from n cubes without overhangs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

JavaScript

Formula

Extensions