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 1-6 of 6 results.

A001933 Number of chessboard polyominoes with n squares.

Original entry on oeis.org

2, 1, 4, 7, 24, 62, 216, 710, 2570, 9215, 34146, 126853, 477182, 1802673, 6853152, 26153758, 100215818, 385226201, 1485248464, 5741275753, 22246121356, 86383454582, 336094015456, 1309998396933, 5114454089528, 19998173763831, 78306021876974, 307022186132259, 1205243906123956, 4736694016531135
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Chessboard-colored polyominoes, considering to be distinct two shapes that cannot be mapped onto each other by any form of symmetry. For example, there are two distinct monominoes, one black, one white. There is only one domino, with one black square, and one white. - John Mason, Nov 25 2013

References

  • W. F. Lunnon, personal communication.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001071, A000105, A121198, A234006 (free polyominoes of size 2n that have at least reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares), A234007 (free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry), A234008 (free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry).

Formula

For odd n, a(n) = 2*A000105(n).
For n multiple of 2 but not of 4, a(n) = 2*A000105(n) - (A234006(n/2) + A234008(n/2)).
For n multiple of 4, a(n) = 2*A000105(n) - (A234006(n/2) + A234008(n/2) + A234007(n/4)). - John Mason, Dec 23 2021

Extensions

a(14)-a(17) from Joseph Myers, Oct 01 2011
a(18)-a(23) from John Mason, Dec 05 2013
a(24)-a(30) from John Mason, Dec 23 2021

A234007 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner, but not having reflective symmetry.

Original entry on oeis.org

0, 1, 2, 9, 30, 110, 387, 1419, 5185, 19225, 71634, 269250, 1017260, 3864267, 14742260, 56470053, 217052829, 836878982
Offset: 1

Views

Author

John Mason, Dec 18 2013

Keywords

Comments

The number of free polyominoes of size 4n that have 90-degree rotational symmetry about a point that coincides with the corner of a square, and that have not at the same time reflective symmetry. Note that for polyominoes which have a hole in the center, the center of rotation will be the corner of a square within the hole, rather than being the corner of a square of the polyomino itself. The sequence is defined for 4n rather than n as polyominoes of size not a multiple of 4 cannot have the required symmetry.
The sequence enumerates a subset of the polyominoes enumerated by A144553.

Crossrefs

Cf. A000105, A001933, A234006, A234008, A234009, A234010, A144553.

Extensions

a(8)-a(13) from Sean A. Irvine, Jul 04 2019
a(14)-a(18) from John Mason, Feb 02 2022

A234008 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side, but no reflective symmetry.

Original entry on oeis.org

0, 1, 4, 16, 60, 231, 877, 3362, 12905, 49825, 193003, 750581, 2927792, 11453171, 44911853, 176499605, 694954416, 2741031257, 10827727980, 42831355495, 169640762209, 672657218163, 2669990735153, 10608176066076, 42184579054003
Offset: 1

Views

Author

John Mason, Dec 18 2013

Keywords

Comments

The number of free polyominoes of size 2n that have 180-degree rotational symmetry about a point that coincides with the midpoint of a side a square, and that have not at the same time any reflective symmetry. Note that for polyominoes which have a hole in the center, the center of rotation will be the midpoint of a side of a square within the hole, rather than being the midpoint of a side of a square of the polyomino itself. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.
The sequence enumerates a subset of the polyominoes enumerated by A006747.

Crossrefs

Cf. A000105, A001933, A234006, A234007, A234009, A234010, A006747.

Extensions

a(12)-a(18) from John Mason, Dec 13 2021
a(19)-a(25) from John Mason, Apr 15 2023

A234009 Free polyominoes with 4n squares, having 90-degree rotational symmetry about a square corner.

Original entry on oeis.org

1, 1, 4, 10, 35, 114, 403, 1432, 5239, 19271, 71820, 269417, 1017920, 3864879
Offset: 1

Views

Author

John Mason, Dec 18 2013

Keywords

Comments

The number of free polyominoes of size 4n that have 90-degree rotational symmetry about a point that coincides with the corner of a square, independently of any other symmetries. Note that for polyominoes which have a hole in the center, the center of rotation will be the corner of a square within the hole, rather than being the corner of a square of the polyomino itself. The sequence is defined for 4n rather than n as polyominoes of size not multiple of 4 cannot have the required symmetry.

Crossrefs

Cf. A000105, A001933, A234006, A234007, A234008, A234010.

Extensions

a(8)-a(14) from John Mason, Dec 13 2021

A234010 Free polyominoes with 2n squares, having 180-degree rotational symmetry about a square mid-side.

Original entry on oeis.org

1, 2, 6, 19, 67, 241, 901, 3398, 12991, 49958, 193317, 751080, 2928956, 11455059, 44916219, 176506797, 694970938, 2741058805, 10827790934, 42831461499, 169641003412, 672657627655, 2669991663529, 10608177653227, 42184582641002
Offset: 1

Views

Author

John Mason, Dec 18 2013

Keywords

Comments

The number of free polyominoes of size 2n that have 180-degree rotational symmetry about a point that coincides with the midpoint of a side a square, independently of any reflective symmetry. Note that for polyominoes which have a hole in the center, the center of rotation will be the midpoint of a side of a square within the hole, rather than being the midpoint of a side of a square of the polyomino itself. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.

Crossrefs

Cf. A000105, A001933, A234006, A234007, A234008, A234009, A346799.

Formula

a(n) = A346799(n) + A234008(n).

Extensions

More terms from John Mason, Dec 17 2021
More terms from John Mason, Apr 15 2023

A001071 Number of one-sided chessboard polyominoes with n cells.

Original entry on oeis.org

2, 1, 4, 10, 36, 108, 392, 1363, 5000, 18223, 67792, 252938, 952540, 3602478, 13699554, 52296713, 200406388, 770411478, 2970401696, 11482395526, 44491881090, 172766311857, 672186650116
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Two polyominoes cut from a chessboard are considered the same for this sequence if the shapes of the polyominoes are related by a rotation or translation, and the colorings are related by any symmetry including a reflection. - Joseph Myers, Oct 01 2011

References

  • W. F. Lunnon, personal communication.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001933, A121198.

Formula

a(n) = 2*O(n) - M(n) - 2*(R90(n) + R180(n)), where:
O(n)=A000988(n),
for even n, M(n) = A234006(n/2), otherwise 0,
for n multiple of 4, R90(n) = A234007(n/4), otherwise 0,
for even n, R180(n) = A234008(n/2), otherwise 0

Extensions

Extended by Joseph Myers, Oct 01 2011
a(18)-a(23) by John Mason, Jan 02 2014
Showing 1-6 of 6 results.

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

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