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 1-5 of 5 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

Views

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

A121195 Consider the tiling of the plane with squares of two different sizes as in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. a(n) is the number of connected figures that can be formed on this tiling, from n big squares and n small squares.

Original entry on oeis.org

1, 12, 152, 2538, 45084, 851717
Offset: 1

Views

Author

N. J. A. Sloane, Aug 17 2006

Keywords

Comments

The Zucca web site calls these figures "n-BiSquares".

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Links

Crossrefs

Cf. A121196, A121197, A121198.

Extensions

More terms from Don Reble, Aug 17 2007

A121197 Consider the tiling of the plane with squares of two different sizes as in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. a(n) is the number of connected figures that can be formed on this tiling, from any n squares.

Original entry on oeis.org

2, 2, 8, 34, 158, 777, 4006, 21224, 114348, 624222, 3441050, 19121530, 106957272
Offset: 1

Views

Author

N. J. A. Sloane, Aug 17 2006

Keywords

Comments

The Zucca web site calls these figures "n-DifferentSquares".
Also the number of one-sided polyforms on the faces of the truncated square tiling. - Peter Kagey, May 24 2025

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Links

Crossrefs

Cf. A121195, A121196, A121198, A343577.

Extensions

More terms from Don Reble, Aug 17 2007
a(13) from Joseph Myers, Oct 06 2011

A121196 Consider the tiling of the plane with squares of two different sizes as in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. a(n) is the number of connected figures that can be formed on this tiling, from n tiles, each composed of a big square and an adjacent little square.

Original entry on oeis.org

1, 11, 114, 1519, 20769
Offset: 1

Views

Author

N. J. A. Sloane, Aug 17 2006

Keywords

Comments

The Zucca web site calls these figures "n-PairSquares".

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Links

Crossrefs

Cf. A121195, A121197, A121198.

Extensions

Better definition from Don Reble, Aug 17 2007

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 1-5 of 5 results.

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

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