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.

Previous Showing 11-20 of 21 results. Next

A151525 Number of poly-IH64-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 2, 4, 12, 35, 116, 392, 1390, 4998, 18321, 67791, 253288, 952527, 3603761, 13699516, 52301427, 200406183, 770429000, 2970400815, 11482461055, 44491876993, 172766558719, 672186631950, 2619995431640, 10228902801505, 39996342220199, 156612023001490, 614044351536722
Offset: 1

Views

Author

Ed Pegg Jr, May 13 2009

Keywords

Comments

Equivalently, polyominoes where two polyominoes are considered the same if and only if they are related by a translation or a reflection in a horizontal line. Formerly described as one-sided polyrects, but that is A151522.

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 6.2 and 9.4.

Links

Crossrefs

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

Formula

a(n) = 4*A006749(n) + 3*A006746(n) + 2*A006748(n) + 2*A006747(n) + 2*A056877(n) + A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Extensions

Edited and a(13)-a(18) by Joseph Myers, Nov 24 2010
a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A182645 Number of poly-IH68-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 4, 10, 34, 110, 388, 1369, 4982, 18246, 67727, 253014, 952275, 3602743, 13698525, 52297602, 200402285, 770414503, 2970385477, 11482405741, 44491816601, 172766346508, 672186393972, 2619994613794, 10228901862928, 39996339056273, 156612019296546, 614044339256951
Offset: 1

Views

Author

Joseph Myers, Nov 24 2010

Keywords

Comments

Equivalently, polyominoes where two polyominoes are considered the same if and only if they are related by a translation or a reflection in a NE-SW diagonal line.

References

  • Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 6.2 and 9.4.

Links

Crossrefs

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

Formula

a(n) = 4*A006749(n) + 2*A006746(n) + 3*A006748(n) + 2*A006747(n) + A056877(n) + 2*A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Extensions

a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A210997 Number of free polyominoes with 2n-1 cells.

Original entry on oeis.org

1, 2, 12, 108, 1285, 17073, 238591, 3426576, 50107909, 742624232, 11123060678, 168047007728, 2557227044764, 39153010938487, 602621953061978, 9317706529987950, 144648268175306702, 2253491528465905342, 35218318816847951974
Offset: 1

Views

Author

Omar E. Pol, Sep 15 2012

Keywords

Links

Crossrefs

Bisection of A000105.
Cf. A000988, A210989, A210996.

A210989 Number of one-sided polyominoes with 2n-1 cells.

Original entry on oeis.org

1, 2, 18, 196, 2500, 33896, 476270, 6849777, 100203194, 1485200848, 22245940545, 336093325058, 5114451441106, 78306011677182, 1205243866707468, 18635412907198670, 289296535756895985, 4506983054619138245, 70436637624668665265
Offset: 1

Views

Author

Omar E. Pol, Sep 16 2012

Keywords

Links

Crossrefs

Bisection of A000988.
Cf. A210988, A210997.

A348848 Number of oriented polyominoes with 4n cells that have fourfold rotational symmetry centered at a vertex.

Original entry on oeis.org

1, 2, 6, 19, 65, 224, 790, 2851, 10424, 38496, 143454, 538667, 2035180, 7729146, 29486904, 112942373, 434114384, 1673766428, 6471199322, 25081542410, 97431694571, 379256586232, 1479022885116
Offset: 1

Views

Author

Robert A. Russell, Nov 01 2021

Keywords

Comments

These are polyominoes of the regular tiling with Schläfli symbol {4,4}. For oriented polyominoes, chiral pairs are counted as two. This is one of the five sequences, along with A001168, needed to calculate the number of oriented polyominoes, A000988. It is the C90(n/4) sequence in the Shirakawa link. The calculation follows Redelmeier's method of inner rings.

Examples

			For a(1)=1, the polyomino is a 2 X 2 square. For a(2)=2, the two polyominoes are a chiral pair having a central 2 X 2 square with one cell attached to each edge of that square.

Links

Crossrefs

Cf. A000988, A144553, A348849 (cell center).
Inner rings: A324406, A324407, A324408, A324409.

A348849 Number of fixed polyominoes with n cells that have fourfold rotational symmetry centered at the center of a cell.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 3, 6, 0, 0, 10, 18, 0, 0, 35, 57, 0, 0, 126, 191, 0, 0, 461, 658, 0, 0, 1699, 2308, 0, 0, 6315, 8241, 0, 0, 23686, 29853, 0, 0, 89432, 109268, 0, 0, 339473, 403450, 0, 0, 1294826, 1501074
Offset: 1

Views

Author

Robert A. Russell, Nov 01 2021

Keywords

Comments

These are polyominoes of the regular tiling with Schläfli symbol {4,4}. Chiral pairs are counted as two. This is one of the five sequences, along with A001168, needed to calculate the number of oriented polyominoes, A000988. It is the F90 sequence in the Shirakawa link. The calculation follows Redelmeier's method of determining inner rings.

Examples

			For a(9)=2, the polyomino is a 3 X 3 square or a row and column of five cells sharing their central cells.

Links

Crossrefs

Cf. A000988, A144553, A348848 (vertex center).
Inner rings: A324406, A324407, A324408, A324409.

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

A354306 Number of one-sided polypentagrams with n cells.

Original entry on oeis.org

1, 2, 7, 62, 459, 4040, 35386, 321639, 2958100, 27585931, 259670736
Offset: 1

Views

Author

Aaron N. Siegel, May 23 2022

Keywords

Comments

Polypentagrams are defined in A211179. "One-sided" means that mirror images are considered distinct.

Links

Crossrefs

Cf. A211179, A103465, A000105, A000988.

A100632 Number of Shapes(n, d) for a given number of polyhypercubes / polytopominoes n in a given dimensional space d.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 7, 8, 7, 1, 18, 29, 27, 26, 1, 60, 166
Offset: 1

Views

Author

Jeffrey C. Jacobs (timehorse(AT)starship.python.net), Dec 03 2004

Keywords

Comments

a(1/2 * n * (n-1) + d) gives values of Shapes(n, d) for n > 0, d > 0, d <= n. For d > n, use a(1/2 * n * (n + 1)) or A005519's a(n).
A polytopomino shape is a shape constructed of n contiguous hypercubes that is invariant over rotation but not necessarily over 'flipping', i.e. mirror images are distinct. See example for details of when flipping is or is not considered.
A000988 gives values for Shape(n, 2), e.g. a(1/2 * n * (n - 1) + 2) and A000162 for Shape(n, 3), e.g. a(1/2 * n * (n - 1) + 3.
A005519 gives values for Shape(n, n), e.g. a(1/2 * n * (n + 1)). These shapes always can be expressed using only n-1 dimensions and therefore contain no mirror-image or "flipped" shapes.
Shape(n, d) is the union of all shapes with n points that can be expressed in a dimension x for x from 1 to d - 1 where "flipped" shapes are excluded plus all shapes with n points that must be expressed by at least d dimensions where "flipped" shapes are included. A049429 gives values for x in [1, d - 1] and n.
Specificly, if b(m) defines the sequence A049429, b(1/2 * n * (n + 1) + x) is the term for all shapes that must be expressed in at least x dimensions and containing n points.
There is no sequence that describes shapes with n points and exactly d dimensions for which "flipped" shapes are considered distinct, so this formula cannot be completely expressed as the sum of other formula.
The main difficulty in computing this sequence is in a) the fast implemention of a set (as in a set of points [shape] and a set of shapes [Shape(n, d)]), especially with respect to rotation of points and b) the difficulty in eliminating duplicate entries.
The later case is difficult because in order to determine whether two shapes are the same, one must compute all possible R in order to determine the R that may orient shape X the same as shape Y. The translation vector T is uniquely given based on R but requires finding the minimum point of the bounding hypercube of each shape that is linear with respect to d.
Ideally, a good algorithm for b must be found, especially if a "definitive orientation" can be determined such that all shapes will be oriented using the definitive orientation before being compared and thus the comparison consists only of comparing the points in X and Y to make sure they are the same.
Also, it should be possible to reduce the loop over Cardinal Vectors since some vectors are equivalent, such as adding (1, 0) or (-1, 0) to the point (0, 0) since the shape has symmetry and therefore both new shapes are equivalent.

Examples

			Example 1: a(9) gives the number of shapes in Shape(4, 3). We describe these 3-dimensional shapes by using 2 rows of text where "O" represents a block in the z=0 plane and "2" two stacked blocks, the first in the z=0 plane, the second in the z=1 plane.
Shape(4, 3) consists of
OOOO ..0 .0. 00 00. 0. 0. .0
.... 000 000 00 .00 20 02 20
The second shape is considered to be the same as
OOO
..O
because it can be expressed in 2 dimensions and we are allowed 3 (d = 3) so these two shapes are the same despite being flipped. However the last two shapes require 3 dimensions to express and because that is equal to or greater than d = 3, the flipped shapes are considered distinct.
This is equivalent to saying that in 3 dimensions there is no physical way to turn or move the second to last shape to make it look like the last.
Example 2: a(8) gives the number of shapes in Shape(4, 2). This is equivalent to the set of 1-sided polyominoes consisting of 4 squares.

Links

Crossrefs

Cf. A005519, A000988, A000162.

Formula

Let a shape consist of a set of n integral points such that all points are adjacent to at least 1 other point and that all points are connected either directly or indirectly through adjacency.
Let two points be adjacent if and only if the distance between point A and point B is given by a unit vector which lies parallel to one of the Cartesian axes in d dimensional space.
E.g. if d is 2 and n is 2, a shape may consist of the points (0, 0) and (1, 0). The distance between these points would then be the unit vector (1, 0) which lies parallel to the x-axis.
Two shapes, X and Y are considered the same if and only if there exists some rotation unit matrix R and some translation vector T for which the set of points X * R + T is equal to the set of points Y. The unit rotation R must have determinant 1.
A determinant of -1 for R is considered a "flip" and is therefore not allowed. However it should be noted that there will always exist an R[d+1] such that R[d+1] = [[R 0] [0 det(R)]], which always has a determinant of 1.
Thus when considering a higher-order dimension, a flip in a lower dimension is now possible. In other words, Shapes are 1-sided only if they must be represented using at least d dimensions.
The Set of Cardinal Vectors consists of all unit vectors parallel to a Cartesian axis for the given dimension d. Thus when d is 2, the Set of Cardinal Vectors consists of { (1, 0) (0, 1) (-1, 0) (0, -1) }.
We then define Shape(n, d) recursively as follows:
Shape(1, d) consists of the single set containing a single point (0, 0, ..., 0) in d-Space, e.g. Shape(1, d) = { { (0, 0, ..., 0) } } for all d.
Shape(n+1, d) consists of all shapes generated by:
For each shape S in Shape(n, d):
For each point P in S:
For each vector V in the Set of Cardinal Vectors:
If P + V is not in S:
Shape(n+1, d) contains the Shape consisting of the union of S and { (P + V) }
a(1/2 n * (n - 1) + d) = the number of shapes in the set Shape(n, d).

Extensions

Link updated by William Rex Marshall, Dec 16 2009

A127560 Number of fixed r-celled polyominoes with smallest containing rectangle measuring k by m, read in order r=A056556(n)+1, k=A056560(n)+1, m=A056558(n)+1.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 1, 0, 1, 8, 8, 1, 0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 1, 12, 25, 12, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 18, 44, 18, 0, 1, 16, 50, 50, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 32, 8, 0, 0, 38, 155, 155, 38, 0, 1, 20, 82, 120, 82, 20, 1
Offset: 0

Views

Author

Graeme McRae, Jan 18 2007

Keywords

Comments

The sum of each triangle, i.e. for a given r the sum of a(n) for all n such that r=A056556(n)+1 is the number of r-celled fixed polyominoes, A001168(r).

Examples

			The 5th triangle of the sequence, the number of fixed pentominoes by dimension, is
0,0,0,0,1
0,0,6,12
0,6,25
0,12
1
This indicates, for example, there are 25 fixed pentominos that fit in a 3 X 3 rectangle and 12 fixed pentominos that fit in a 4 X 2 rectangle.

Crossrefs

Cf. A000105, A000988, A001168 Indices for reading by triangles given by A056556, A056560, A056558.
Previous Showing 11-20 of 21 results. Next

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

Content is available under The OEIS End-User License Agreement.