A054247 -id:A054247 - cp's OEIS frontend

A054407 Number of asymmetric n X n binary matrices under action of dihedral group of the square D_4.

0, 0, 0, 36, 7860, 4177152, 8589313152, 70368605798400, 2305842990423490560, 302231454885790246502400, 158456325028519245775644917760, 332306998946228931044233275859009536, 2787593149816327892614636032925171439370240

Offset: 0

Vladeta Jovovic, May 08 2000

Colin Barker, Table of n, a(n) for n = 0..50

Cf. A054247.

(1/8)*(2^(n^2)-2*2^((n^2+n)/2)-3*2^(n^2/2)+2*2^(n^2/4)+2*2^((n^2+2*n)/4)) if n is even and (1/8)*(2^(n^2)-4*2^((n^2+n)/2)-2^((n^2+1)/2)+4*2^((n^2+2*n+1)/4)) if n is odd. - Corrected by Colin Barker, Oct 25 2016

More terms from Colin Barker, Oct 25 2016

A286394 Number of inequivalent n X n matrices over GF(8) under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 8, 666, 16912512, 35184646816768, 4722366500530551259136, 40564819207305653446303190876160, 22300745198530623151211847196048401987796992, 784637716923335095479473759060307277562325323313332617216

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 08 2017

nonn

Burnside's orbit-counting lemma.

María Merino, Table of n, a(n) for n = 0..33
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Column k=8 of A343097.

Cf. A054247, A054739, A054751, A054752, A286392, A286393.

a(n) = (1/8)*(8^(n^2) + 2*8^(n^2/4) + 3*8^(n^2/2) + 2*8^((n^2 + n)/2)) if n is even;

a(n) = (1/8)*(8^(n^2) + 2*8^((n^2 + 3)/4) + 8^((n^2 + 1)/2) + 4*8^((n^2 +n)/2)) if n is odd.

A367522 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under both horizontal and vertical reflection, but not diagonal reflection.

Original entry on oeis.org

1, 4, 84, 8292, 4203520, 8590033024, 70368815480832, 2305843010824323072, 302231454912728264605696, 158456325028529097399561355264, 332306998946228986960926214931349504, 2787593149816327892693735671512138485071872, 93536104789177786765036453099565034406633831137280

Offset: 1

Peter Kagey, Nov 21 2023

nonn

The a(2) = 4 tilings are
  - -   - -   - |       - |
  - -,  | -,  - |, and  | -.

Michael De Vlieger, Table of n, a(n) for n = 1..57
Peter Kagey, Illustration of a(3)=84.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, p. A-6.

Cf. A054247, A295229, A302484, A367523, A367524, A367525.

a[n_] := If[EvenQ[n], 2^(#^2 - 3)*(2 + 3*2^#^2 + 8^#^2) &[n/2], 4^(#^2 - 2 # - 1)*(4^# + 4^#^2 + 8^#) &[(n + 1)/2]]; Array[a, 13] (* Michael De Vlieger, Jul 06 2024 *)

a(2m-1) = 4^(m^2 - 2m - 1)*(4^m + 4^m^2 + 8^m).

a(2m) = 2^(m^2 - 3)*(2 + 3*2^m^2 + 8^m^2).

A367523 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under 90-degree rotations, but not reflections.

Original entry on oeis.org

1, 4, 70, 8292, 4195360, 8590033024, 70368748374016, 2305843010824323072, 302231454903932172107776, 158456325028529097399561355264, 332306998946228968514182141758668800, 2787593149816327892693735671512138485071872, 93536104789177786765035834129545391718695404830720

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(3)=70
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-6, A-7.

Cf. A054247, A295229, A302484, A367522, A367524, A367525.

Table[{2^(-2 + (-4 + n) n) (2^(n (2 + n)) + 2^(1 + 3 n) + 8^n^2), 2^(-3 + n^2) (2 + 3 2^n^2 + 8^n^2)}, {n, 1, 5}] // Flatten

a(2m-1) = 2^(m^2 - 4m - 2)*(2^(3m+1) + 2^(m^2+2m) + 8^m^2).

a(2m) = 2^(m^2 - 3)*(2 + 3*2^m^2 + 8^m^2) = A367522(2m).

A268339 Number of polyominoes with width and height equal to n that are invariant under all symmetries of the square.

Original entry on oeis.org

1, 1, 3, 3, 17, 17, 163, 163, 2753, 2753, 84731, 84731, 4879497, 4879497, 535376723, 535376723, 112921823249, 112921823249, 45931435159067, 45931435159067, 36048888105745113, 36048888105745113, 54568015172025197171, 54568015172025197171, 159197415409641803530753, 159197415409641803530753

Offset: 1

Craig Knecht, Feb 02 2016

nonn

Percolation theory focuses on patterns that provide connectivity. Polyominoes that connect all boundaries of a square are in the percolation neighborhood. This subclass of symmetric polyominoes distinguishes itself for its beauty and its unusual enumeration pattern.

			The ones in this example provide the connective pattern that joins all boundaries of the square.
0 1 1 1 0
1 0 1 0 1
1 1 1 1 1
1 0 1 0 1
0 1 1 1 0

Craig Knecht, Change of state - math becomes art
Craig Knecht, Polyominoe enumeration
Wikipedia, Connective polyominoes with 4 sym-axis
Wikipedia, Water Retention on Mathematical Surfaces

Cf. A054247 (all unique water retention patterns for an n X n square), A268311 (polyominoes that connect all boundaries on a square), A268758.

a(2*n) = a(2*n-1) = A268758(n). - Andrew Howroyd, May 03 2020

Terms a(17) and beyond from Andrew Howroyd, May 03 2020

A268758 Number of polyominoes with width and height equal to 2n that are invariant under all symmetries of the square.

Original entry on oeis.org

1, 3, 17, 163, 2753, 84731, 4879497, 535376723, 112921823249, 45931435159067, 36048888105745113, 54568015172025197171, 159197415409641803530753, 894444473815989281612355579, 9671160618112663336510127727593, 201110001346886305066013828873025811

Offset: 1

Craig Knecht, Feb 14 2016

nonn

Also number of polyominoes with width and height equal to 2n - 1 that are invariant under all symmetries of the square.
Bisection of A268339.
The water retention model for mathematical surfaces is described in the link below. The definition of a "lake" in this model is related to a class of polyominoes in A268339. Percolation theory may refer to these structures as "clusters that touch all boundaries."
Transportation across the square lattice requires a path of continuous edge connected cells. Is a pattern that only connects two opposite boundaries of the square ranked differently from one that connects all four boundaries?
This sequence is part of a effort to classify water retention patterns in a square by their symmetry, their capacity to connect boundaries of the square and the number of edge cells that are connected across opposite boundaries.

			For a(2) = 3: the three polyominoes of width and height 2*2 - 1 = 3 and the corresponding three polynomial of width and height 2*2 = 4 are shown below. Note that each even-dimension polyomino is produced by duplicating the center row/column of an odd-dimension polyomino.
3 X 3:
   0 1 0     1 1 1     1 1 1
   1 1 1     1 0 1     1 1 1
   0 1 0     1 1 1     1 1 1
4 X 4:
  0 1 1 0   1 1 1 1   1 1 1 1
  1 1 1 1   1 0 0 1   1 1 1 1
  1 1 1 1   1 0 0 1   1 1 1 1
  0 1 1 0   1 1 1 1   1 1 1 1

Andrew Howroyd, Table of n, a(n) for n = 1..20
Craig Knecht, Connections between boundaries of the square
Craig Knecht Retention capacity of a random surface, arXiv:1110.6166 [cond-mat.dis-nn], 2011-2012.
Wikipedia, Water Retention on Mathematical Surfaces

Cf. A268339, A268404, A331878.

a(n) = A331878(n) - 3*A331878(n-1) + 3*A331878(n-2) - A331878(n-3) for n >= 4. - Andrew Howroyd, May 03 2020

Terms a(9) and beyond from Andrew Howroyd, May 03 2020

A286396 Number of inequivalent n X n matrices over GF(9) under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 9, 1035, 48700845, 231628411446741, 89737248564744874067889, 2816049943117424212512789695666175, 7158021121277935153545945911617993395398302485, 1473773072217322896440109113309952350877179744639518847951721

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 08 2017

nonn

Burnside's orbit-counting lemma.

María Merino, Table of n, a(n) for n = 0..32
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Column k=9 of A343097.

Cf. A054247, A054739, A054751, A054752, A286392, A286393, A286394.

Table[1/8*(9^(n^2) + 2*9^((n^2 + 3 #)/4) + (3 - 2 #)*9^((n^2 + #)/2) + (2 + 2 #)*9^((n^2 + n)/2)) &@ Boole@ OddQ@ n, {n, 0, 7}] (* Michael De Vlieger, May 12 2017 *)

a(n) = (1/8)*(9^(n^2) + 2*9^(n^2/4) + 3*9^(n^2/2) + 2*9^((n^2 + n)/2)) if n is even;

a(n) = (1/8)*(9^(n^2) + 2*9^((n^2 + 3)/4) + 9^((n^2 + 1)/2) + 4*9^((n^2 + n)/2)) if n is odd.

A286397 Number of inequivalent n X n matrices over an alphabet of size 10 under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 10, 1540, 125512750, 1250002537502500, 1250000000501250002500000, 125000000000000250375000000250000000, 1250000000000000000005001250000000002500000000000

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 08 2017

nonn

Burnside's orbit-counting lemma.

María Merino, Table of n, a(n) for n = 0..20
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Column k=10 of A343097.

Cf. A054247, A054739, A054751, A054752, A286392, A286393, A286394, A286395.

Table[1/8*(10^(n^2) + 2*10^((n^2 + 3 #)/4) + (3 - 2 #)*10^((n^2 + #)/2) + (2 + 2 #)*10^((n^2 + n)/2)) &@ Boole@ OddQ@ n, {n, 7}] (* Michael De Vlieger, May 12 2017 *)

a(n) = (1/8)*(10^(n^2) + 2*10^(n^2/4) + 3*10^(n^2/2) + 2*10^((n^2 + n)/2)) if n is even;

a(n) = (1/8)*(10^(n^2) + 2*10^((n^2 + 3)/4) + 10^((n^2 + 1)/2) + 4*10^((n^2 + n)/2)) if n is odd.

A261798 Maximum water retention of an associative magic square of order n.

Original entry on oeis.org

0, 0, 0, 15, 59, 0, 361, 704, 1247, 0

Offset: 1

Craig Knecht, Sep 01 2015

Two of the most famous magic squares are associative magic squares - the Lo Shu magic square and Dürer's magic square. Al Zimmermann's programming contest in 2010 produced the presently known maximum retention values for magic squares order 4 to 28 A201126. No concerted effort has been made to find the maximum retention for associative magic squares.
There are 4211744 different water retention patterns for a 7 x 7 square A054247 and 1.12*10^18 different order 7 associative magic squares. There is no proof that the presently stated maximum retention values greater than order 5 are actually the maximum possible retention.
a(11) >= 3226, a(12) >= 4840, a(13) >= 6972.
The Wikipedia link below shows the first attempt to classify a set of data by its water retention. Here the 48 associative order 4 magic squares are thus classified. Perhaps there might be some correlation between this surface evaluation and Mohs hardness scale.

			(16  3  2  13)
(5  10 11   8)
(9   6  7  12)
(4  15  14  1)
This is Albrecht Dürer's famous magic square in Melancholia I. Dürer put the date of its creation (1514) in the numbers in the bottom row. This square holds 5 units of water.

Craig Knecht, Order 5 associative magic square.
Craig Knecht, Order 7 associative magic square.
Craig Knecht, Order 8 associative magic square.
Craig Knecht, Order 9 associative magic square.
Craig Knecht, Order 12 associative magic square.
Johan Ofverstedt, Water Retention on Magic Squares with Constraint Based Local Search.
Wikipedia, Listing by water retention capacity. and Water retention on mathematical surfaces.

Cf. A201126 (water retention on magic squares), A201127 (water retention on semi-magic squares), A261347 (water retention on number squares).

A285766 Maximum spillway height for a zero or one bend minimal area lake in a number square.

Original entry on oeis.org

0, 0, 6, 10, 15, 22, 31, 42, 55, 70, 87, 106, 127, 150, 175, 202, 231, 262, 295, 330, 367, 406, 447, 490, 535, 582, 631, 682, 735, 790, 847, 906, 967, 1030, 1095, 1162, 1231, 1302, 1375, 1450, 1527, 1606, 1687, 1770, 1855, 1942, 2031, 2122, 2215, 2310, 2407

Offset: 0

Craig Knecht, May 04 2017

nonn

The water retention model for mathematical surfaces led to definitions for a lake and a pond.  These lakes and ponds divide the square up in interesting ways. This sequence looks at the spillway heights in zero or one bend minimal area lakes.
A lake has dimensions of (n-2) X (n-2) when the square is n X n.  All other water retaining areas are ponds.
A number square contains the numbers 1 to n^2 without repeats.
The larger terms are a(n)= n^2+6 or A114949.

			For the 4 X 4 square a example of a smallest lake is shown. The values 1,2,3 form the lake. The pathway of least resistance off the square is the spillway value 10.
   ( 4  16  15   5)
   (10   1   2  14)
   ( 6  11   3  13)
   ( 7   8  12   9)

Craig Knecht, Minimal lake types in a 7x7 square.
Craig Knecht, Minimal lake area in a square
Wikipedia, Water retention on mathematical surfaces

Cf. A054247, A201126, A268311.

Conjectures from Colin Barker, May 07 2017: (Start)
G.f.: x^2*(6 - 8*x + 3*x^2 + x^3) / (1 - x)^3.
a(n) = 7 - 2*n + n^2 for n>2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.
(End)

cp's OEIS Frontend

A054407 Number of asymmetric n X n binary matrices under action of dihedral group of the square D_4.

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A286394 Number of inequivalent n X n matrices over GF(8) under action of dihedral group of the square D_4.

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A367522 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under both horizontal and vertical reflection, but not diagonal reflection.

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A367523 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under 90-degree rotations, but not reflections.

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A268339 Number of polyominoes with width and height equal to n that are invariant under all symmetries of the square.

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A268758 Number of polyominoes with width and height equal to 2n that are invariant under all symmetries of the square.

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A286396 Number of inequivalent n X n matrices over GF(9) under action of dihedral group of the square D_4.

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A286397 Number of inequivalent n X n matrices over an alphabet of size 10 under action of dihedral group of the square D_4.

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A261798 Maximum water retention of an associative magic square of order n.

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