A054739 -id:A054739 - cp's OEIS frontend

A343097 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 21, 102, 1, 0, 1, 5, 55, 2862, 8548, 1, 0, 1, 6, 120, 34960, 5398083, 4211744, 1, 0, 1, 7, 231, 252375, 537157696, 105918450471, 8590557312, 1, 0, 1, 8, 406, 1284066, 19076074375, 140738033618944, 18761832172500795, 70368882591744, 1, 0

Offset: 0

Andrew Howroyd, Apr 14 2021

			Array begins:
====================================================================
n\k | 0 1       2            3               4                 5
----+---------------------------------------------------------------
  0 | 1 1       1            1               1                 1 ...
  1 | 0 1       2            3               4                 5 ...
  2 | 0 1       6           21              55               120 ...
  3 | 0 1     102         2862           34960            252375 ...
  4 | 0 1    8548      5398083       537157696       19076074375 ...
  5 | 0 1 4211744 105918450471 140738033618944 37252918396015625 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..860

Rows 0..5 are A000012, A001477, A002817, A217331, A217338, A283033.
Columns 0..10 are A000007, A000012, A054247, A054739, A054751, A054752, A286392, A286393, A286394, A286396, A286397.
Main diagonal is A001326.
Cf. A246106, A343095, A343875.

T(n,k) = {(k^(n^2) + 2*k^((n^2 + 3*(n%2))/4) + k^((n^2 + (n%2))/2) + 2*k^(n*(n+1)/2) + 2*k^(n*(n+n%2)/2) )/8}

T(n,k) = (k^(n^2) + 2*k^((n^2 + 3*(n mod 2))/4) + k^((n^2 + (n mod 2))/2) + 2*k^(n*(n+1)/2) + 2*k^(n*(n + n mod 2)/2) )/8.

A286447 Number of inequivalent n X n matrices over GF(3) under action of dihedral group of the square D_4, with a third of 1's, 2's and 3's (ordered occurrences rounded up/down if n^2 != 0 mod 3).

Original entry on oeis.org

1, 1, 2, 228, 252642, 3286762710, 423091508279496, 488322998306377824150, 5405955851967092442258037800, 561273297862912365721571649672300480, 524055990531978935668322776302483856990581000

Offset: 0

María Merino, Imanol Unanue, May 11 2017

nonn

			For n=3 the a(3)=228 solutions are colorings of 3 X 3 matrices in 3 colors inequivalent under the action of D_4 with exactly 3 occurrences of each color (coefficient of x1^3 x2^3 x3^3).

María Merino, Table of n, a(n) for n = 0..45
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A054739, A082963.

G.f.: g(x1,x2,x3) = 1/8*(y1^(n^2) + 2*y1^n*y2^((n^2 - n)/2) + 3*y2^(n^2/2) + 2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2) + 4*y1^n*y2^((n^2 - n)/2) + y1*y2^((n^2 - 1)/2) + 2*y1*y4^((n^2 - 1)/4)) if n odd, where coefficient correspond to y1 = x1 + x2 + x3, y2 = x1^2 + x2^2 + x3^2, y4 = x1^4 + x2^4 + x3^4 and occurrences of numbers are ceiling(n^2/3) for 1's and floor(n^2/3) for 2's and 3's.

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.

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.

A047938 Number of ways to label points of an n X n grid with 3 colors, up to rotational symmetry.

Original entry on oeis.org

1, 3, 24, 4995, 10763361, 211822552035, 37523658921114744, 59824832307866205347043, 858420955073128584419531008641, 110856622060759442496180656754310346403, 128844380183002832759115461915902241562318377784, 1347757724935823407809884872163997148505019182125807296675

Offset: 0

Rob Pratt

nonn

Labels of adjacent points need not be distinct.

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

Column k=3 of A343095.
See A068253 for number of proper colorings.
Cf. A054739, A047937, A047939.

a(n) = (m^(n^2) + 2 m^((n^2 + 3 (n mod 2))/4) + m^((n^2 + (n mod 2))/2))/4, with m = 3.

Terms a(10) and beyond from Andrew Howroyd, Apr 14 2021

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.

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.

cp's OEIS Frontend

A343097 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.

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A286447 Number of inequivalent n X n matrices over GF(3) under action of dihedral group of the square D_4, with a third of 1's, 2's and 3's (ordered occurrences rounded up/down if n^2 != 0 mod 3).

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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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Mathematica

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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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A047938 Number of ways to label points of an n X n grid with 3 colors, up to rotational symmetry.

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Extensions

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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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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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

Formula

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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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

Formula