A054247 -id:A054247 - cp's OEIS frontend

A054751 Number of inequivalent n X n matrices over GF(4) under action of dihedral group of the square D_4.

1, 4, 55, 34960, 537157696, 140738033618944, 590295811483987148800, 39614081257168338331296071680, 42535295865117309120430975675097153536, 730750818665451459102461990840694008379514814464, 200867255532373784442745261867639247948787687313041365401600

Offset: 0

Vladeta Jovovic, May 15 2000

Andrew Howroyd, Table of n, a(n) for n = 0..25

Column k=4 of A343097.

Cf. A054247.

Table[If[EvenQ[n],(4^n^2+2*4^(n^2/4)+3*4^(n^2/2)+2*4^((n^2+n)/2))/8,(4^n^2+2*4^((n^2+3)/4)+4^((n^2+1)/2)+4*4^((n^2+n)/2))/8],{n,0,10}] (* Harvey P. Dale, Aug 16 2021 *)

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

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

Terms a(10) and beyond from Andrew Howroyd, Apr 15 2021

A054752 Number of inequivalent n X n matrices over GF(5) under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 5, 120, 252375, 19076074375, 37252918396015625, 1818989403666496277343750, 2220446049250331744551658935546875, 67762635780344027129112510010600128173828125, 51698788284564229679463057470911735435947895050048828125

Offset: 0

Vladeta Jovovic, May 15 2000

Andrew Howroyd, Table of n, a(n) for n = 0..25

Column k=5 of A343097.

Cf. A054247.

a(n) = 1/8*(5^(n^2) + 2*5^(n^2/4) + 3*5^(n^2/2) + 2*5^((n^2+n)/2)) if n is even;

a(n) = 1/8*(5^(n^2) + 2*5^((n^2+3)/4) + 5^((n^2+1)/2) + 4*5^((n^2+n)/2)) if n is odd.

Terms a(9) and beyond from Andrew Howroyd, Apr 15 2021

A367524 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 horizontal reflection, but no other symmetries of the square.

Original entry on oeis.org

1, 39, 32896, 536895552, 140737496743936, 590295810384475521024, 39614081257132309534260330496, 42535295865117307939839354957685850112, 730750818665451459101843020821051317142553624576, 200867255532373784442745261543120694290360960529885344825344

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Also, this is the number ways of tiling the n X n grid up to the symmetries of the square by a tile that is fixed under 180-degree rotation, but no other symmetries of the square.

Peter Kagey, Illustration of a(2)=39
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, A-8.

Cf. A054247, A295229, A302484, A367522, A367523, A367525.

Table[{2^(4 m^2 - 4 m - 2) (2 + 2^(2 m - 1)^2), 2^(2 m^2 - 3) (2 + 3*4^m^2 + 64^m^2)}, {m, 1, 5}] // Flatten

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

a(2m) = 2^(2m^2 - 3)*(2 + 3*4^m^2 + 64^m^2).

A286392 Number of inequivalent n X n matrices over an alphabet of size 6 under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 6, 231, 1284066, 352654485156, 3553786240466361696, 1289303099816839265917858176, 16839193280515921004090301582258640896, 7917535832871659713272867459049024690729209839616

Offset: 0

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

Computed using Burnside's orbit-counting lemma.

María Merino, Table of n, a(n) for n = 0..35
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Column k=6 of A343097.

Cf. A054247, A054739, A054751, A054752.

Table[1/8*(6^(n^2) + 2*6^((n^2 + 3 #)/4) + (3 - 2 #)*6^((n^2 + #)/2) + (2 + 2 #)*6^((n^2 + n)/2)) &@ Boole[OddQ@ n], {n, 10}] (* Michael De Vlieger, May 08 2017 *)

a(n) = (1/8)*(6^(n^2) + 2*6^(n^2/4) + 3*6^(n^2/2) + 2*6^((n^2 + n)/2)) if n is even;

a(n) = (1/8)*(6^(n^2) + 2*6^((n^2 + 3)/4) + 6^((n^2 + 1)/2) + 4*6^((n^2 + n)/2)) if n is odd.

A367525 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is not fixed under any of the symmetries of the square.

Original entry on oeis.org

1, 538, 16777216, 35184378381312, 4722366482869645213696, 40564819207303347603293977182208, 22300745198530623141535718272648361505980416, 784637716923335095479473677930668862955643627524327473152, 1766847064778384329583297500742918515827483896875618958121606201292619776

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(2)=538
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-8.

Cf. A054247, A295229, A302484, A367522, A367523, A367524.

Table[{4096^(m^2 - m), 8^(m^2 - 1) (512^m^2 + 3*8^m^2 + 2)}, {m, 1, 5}] // Flatten

a(2m-1) = 4096^(m^2 - m).

a(2m) = 8^(m^2 - 1)*(512^m^2 + 3*8^m^2 + 2).

A082966 Number of inequivalent ways (mod D_4) three checkers can be placed on an n X n board.

Original entry on oeis.org

0, 1, 16, 77, 319, 920, 2397, 5278, 10874, 20355, 36390, 61171, 99441, 154882, 235179, 346060, 499172, 702933, 974124, 1324585, 1777555, 2349116, 3070441, 3962762, 5066814, 6409975, 8044322, 10004463, 12355749, 15141190, 18441495, 22309336, 26843016, 32106217

Offset: 1

Vladeta Jovovic, May 27 2003

Erich Friedman, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

Cf. A014409, A054252, A242279, A242358, A054247.

Rest@ CoefficientList[Series[x^2*(x^8 - x^7 - 4 x^6 - 55 x^5 - 56 x^4 - 83 x^3 - 28 x^2 - 13 x - 1)/((x - 1)^7*(x + 1)^4), {x, 0, 34}], x] (* Michael De Vlieger, Oct 03 2016 *)

a(n) = (1/48)*(n-1)*(n^5+n^4-2*n^3+14*n^2-5*n+3) if n is odd;
a(n) = (1/48)*n*(n-1)*(n^2-n+2)*(n^2+2*n-2) if n is even.
G.f.: x^2*(x^8-x^7-4*x^6-55*x^5-56*x^4-83*x^3-28*x^2-13*x-1) / ((x-1)^7*(x+1)^4). - Colin Barker, Jul 11 2013
a(n) = A054247(n, 3) = A054247(n, n^2-3), n >= 1. - Wolfdieter Lang, Oct 03 2016
E.g.f.: (x*(3 + 24*x + 88*x^2 + 62*x^3 + 15*x^4 + x^5)*cosh(x) + (-3 + 39*x^2 + 80*x^3 + 62*x^4 + 15*x^5 + x^6)*sinh(x))/48. - Stefano Spezia, Apr 14 2022

More terms from Colin Barker, Jul 11 2013

A268404 Number of fixed polyominoes that have a width and height of n.

Original entry on oeis.org

1, 5, 111, 7943, 1890403, 1562052227, 4617328590967, 49605487608825311, 1951842619769780119767, 282220061839181920696642671, 150134849621798165832163223922131, 293909551918134914019004192289440616787, 2116817972794640259940977362779552773322908743

Offset: 1

Craig Knecht, Feb 03 2016

nonn

Iwan Jensen originally provided this sequence.
The sequence also describes the water patterns of lakes in the water retention model.
A lake is defined as a body of water with dimensions of n X n when the size of the square is (n+2) X (n+2). All other bodies of water are ponds.
The 3 X 3 square serves as a tutorial for the following three nomenclatures: (1) The total number of distinct water patterns is 102 and includes lakes and ponds. (2) The number of free lake-type polyominoes is 24. (3) The number of fixed lake-type polyominoes is 111. See the explanatory graphics in the link section.
John Mason has looked at free polyominoes in rectangles; see A268371.
Anna Skelt initiated the discussion on the definition of a lake.

			There are many interesting ways to connect all boundaries of the square with the smallest number of edge-joined cells.
  0 0 0 0 1 0
  0 0 0 0 1 1
  0 0 1 1 1 0
  0 0 1 0 0 0
  1 1 1 0 0 0
  0 1 0 0 0 0

Andrew Howroyd, Table of n, a(n) for n = 1..15
Craig Knecht, 4x4 minimal lake area patterns
Craig Knecht, 5x5 minimal lake area patterns
Craig Knecht, 6x6 minimal lake area patterns
Craig Knecht, 7x7 minimal lake area patterns
Craig Knecht, 24 free lake-type polyominoes 3x3
Craig Knecht, Polyominoe enumeration
Craig Knecht, Walter Trump's 111 fixed lake-type polyominoes 3x3
Wikipedia, Water Retention on Mathematical Surfaces

Main diagonal of A292357.

Cf. A054247 (all unique water retention patterns for an n X n square), A268311 (free polyominoes that connect all boundaries on a square), A268339 (lake patterns that are invariant to all transformations).

A292357 = Cases[Import["https://oeis.org/A292357/b292357.txt", "Table"], {, }][[All, 2]];
a[n_] := A292357[[2n^2 - 2n + 1]];
Array[a, 15] (* Jean-François Alcover, Sep 10 2019 *)

a(12)-a(13) from Andrew Howroyd, Oct 02 2017

A286393 Number of inequivalent n X n matrices over GF(7) under action of dihedral group of the square D_4.

Original entry on oeis.org

1, 7, 406, 5105212, 4154189102413, 167633579843887699759, 331466355732596931093508048522, 32115447190132359991237336502881651018804, 152470060954479462517322396167243320349298407119379801

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..34
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Column k=7 of A343097.

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

a(n) = (1/8)*(7^(n^2) + 2*7^(n^2/4) + 3*7^(n^2/2) + 2*7^((n^2 + n)/2)) if n is even;

a(n) = (1/8)*(7^(n^2) + 2*7^((n^2 + 3)/4) + 7^((n^2 + 1)/2) + 4*7^((n^2 + n)/2)) if n is odd.

A295229 Number of tilings of the n X n grid, using diagonal lines to connect the grid points.

Original entry on oeis.org

1, 6, 84, 8548, 4203520, 8590557312, 70368815480832, 2305843028004192256, 302231454912728264605696, 158456325028538104598816096256, 332306998946228986960926214931349504, 2787593149816327892769293535238052808491008

Offset: 1

Peter Kagey, Nov 18 2017

nonn

The grids are counted up to reflection and rotation.

a(n) <= A295223(n).

			For n = 2, the a(2) = 6 tilings are:
//, \/, /\, \\, /\, and \/.
//  //  //  //  \/      /\

Peter Kagey, Table of n, a(n) for n = 1..57
Andrew Howroyd, Derivation of Formula
Peter Kagey, Example of the 6 tilings of the 2 X 2 grid.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.

Cf. A054247, A295223.

Array[(2^(#^2) + 2*2^(# (# + 1)/2) + If[EvenQ@ #, 3*2^(#^2/2) + 2*2^(#^2/4), 2^((#^2 + 1)/2)])/8 &, 12] (* Michael De Vlieger, Apr 12 2018 *)

a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + if(n%2, 2^((n^2+1)/2), 3*2^(n^2/2) + 2*2^(n^2/4)))/8; \\ Andrew Howroyd, Nov 19 2017

From Andrew Howroyd, Nov 19 2017: (Start)
a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 3*2^(n^2/2) + 2*2^(n^2/4)) / 8 for n even.
a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 2^((n^2+1)/2)) / 8 for n odd. (End)

a(5)-a(12) from Andrew Howroyd, Nov 19 2017

A367526 The number of ways of tiling the n X n grid up to diagonal and antidiagonal reflections by two tiles that are each fixed under both of these reflections.

Original entry on oeis.org

2, 9, 168, 16960, 8407040, 17180983296, 140737630961664, 4611686053860868096, 604462909825456529211392, 316912650057075646247661993984, 664613997892457973921852429862699008, 5575186299632655785536225887234636434636800, 187072209578355573530072906199130068813267662274560

Offset: 1

Peter Kagey, Dec 10 2023

nonn

Peter Kagey, Illustration of a(2)=9
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-9.

Cf. A054247, A367526, A367527, A367528, A367529.

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

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

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

cp's OEIS Frontend

A054751 Number of inequivalent n X n matrices over GF(4) under action of dihedral group of the square D_4.

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A054752 Number of inequivalent n X n matrices over GF(5) under action of dihedral group of the square D_4.

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A367524 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 horizontal reflection, but no other symmetries of the square.

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A286392 Number of inequivalent n X n matrices over an alphabet of size 6 under action of dihedral group of the square D_4.

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A367525 The number of ways of tiling the n X n grid up to the symmetries of the square by a tile that is not fixed under any of the symmetries of the square.

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A082966 Number of inequivalent ways (mod D_4) three checkers can be placed on an n X n board.

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A268404 Number of fixed polyominoes that have a width and height of n.

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A286393 Number of inequivalent n X n matrices over GF(7) under action of dihedral group of the square D_4.

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A295229 Number of tilings of the n X n grid, using diagonal lines to connect the grid points.

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