A295223 -id:A295223 - cp's OEIS frontend

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

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

A367533 The number of ways of tiling the n X n torus 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, 18, 733, 170440, 239035502, 1436110601256, 36028815364865610, 3731252530927004638384, 1584563250299991004659308752, 2746338834266357074512496613490144, 19358285762613388346374725943958077888688, 553468075675608205710323628035216140349636855680

Offset: 1

Peter Kagey, Dec 13 2023

nonn

Peter Kagey, Illustration of a(3)=18
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-21, A-22.

Cf. A255016, A295223, A367522.

A367533[n_] := 1/(8 n^2) * (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + If[OddQ[n], n^2*2^((n^2 + 1)/2), n^2/4 (3*2^(n^2/2) + 2^((n^2 + 4)/2))] + 2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (2^(n*n/LCM[2, c]) + 2^((n - 2)*n/LCM[2, c])*2^(2 n/c))]], n*DivisorSum[n, Function[c, EulerPhi[c] (2^((n - 1)*n/LCM[2, c])*2^(n/c))]]] + 2*If[OddQ[n], 0, n^2 2^(n^2/4 - 1)] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/(2d))]]])

A367534 The number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 90-degree rotation but not reflection.

Original entry on oeis.org

1, 4, 14, 613, 168832, 238686222, 1436101016320, 36028798185029194, 3731252529949661491712, 1584563250285579485868500176, 2746338834266355397535763176765440, 19358285762613388144887089341554236250288, 553468075675608205710276014956782089461163991040

Offset: 1

Peter Kagey, Dec 13 2023

nonn

Peter Kagey, Illustration of a(3)=14
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-21, A-22.

Cf. A255016, A295223, A367523, A367533.

A367534[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + If[OddQ[n], n^2 2^((n^2 + 1)/2), n^2/4 (3*2^(n^2/2) + 2^((n^2 + 4)/2))] + 2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[ c] (2^(n*n/LCM[2, c]) + 2^((n - 2)*n/LCM[2, c]) If[OddQ[c], 0, 2^(2 n/c)])]], n*DivisorSum[n, Function[c, EulerPhi[ c] (2^((n - 1)*n/LCM[2, c]) If[OddQ[c], 0, 2^(n/c)])]]] + 2*If[OddQ[n], n^2 2^((n^2 + 3)/4), n^2/2 (2^(n^2/4) + 2^(n^2/4 + 2))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/(2d))]]])

A367535 The number of ways of tiling the n X n torus 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, 16, 3692, 33570410, 5629501212064, 16397105856182791856, 808450637900676611412052288, 664613997892457939442293683754387488, 9021615045252487149405529092893182593313188608, 2008672555323737844427452615613431716686417747867226446336

Offset: 1

Peter Kagey, Dec 13 2023

nonn

Peter Kagey, Illustration of a(2)=16
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-21, A-22.

Cf. A255016, A295223, A367524, A367533, A367534.

A367535[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] +If[OddQ[n], 0, n^2 (3*2^(n^2 - 2) + 2^(n^2/2)) ] +2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (4^(n*n/LCM[2, c]) + 4^((n - 2)*n/LCM[2, c]) If[OddQ[c], 2, 4]^(2 n/c))]], n*DivisorSum[n, Function[c, EulerPhi[c] (4^((n - 1)*n/LCM[2, c]) If[OddQ[c], 2, 4]^(n/c))]]] +n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 0, EvenQ[d], 2^(n^2/d + 1)]]])

A368137 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 180-degree rotation.

Original entry on oeis.org

1, 23, 3776, 33601130, 5629507922944, 16397105889110874288, 808450637900797243544928256, 664613997892457948377435344457451552, 9021615045252487149406066393257455761827823616, 2008672555323737844427452616231411384297679581096869206528

Offset: 1

Peter Kagey, Dec 16 2023

nonn

Peter Kagey, Illustration of a(2)=23
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-23.

Cf. A255016, A295223, A367524, A367533, A367534, A367535, A367536, A368138.

A368137[n_] := 1/(8 n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] + n^2*If[EvenQ[n], 19*2^(n^2 - 2) + 2^(n^2/2), 2^(n^2 + 1)] + n*If[EvenQ[n], DivisorSum[n, Function[d, EulerPhi[ d] (If[EvenQ[d], 2 (2^(n^2/d) + 4^(n^2/d)), 2^(n^2/d)])]], DivisorSum[n, Function[d, EulerPhi[d] (If[EvenQ[d], 2 (2^(n^2/d) + 4^(n^2/d)), 0])]]])

A368138 Number of ways of tiling the n X n torus up to the symmetries of the square by an asymmetric tile.

Original entry on oeis.org

1, 154, 1864192, 2199026796168, 188894659314785812480, 1126800533536206914843196839296, 455117248949604553908892209645884928950272, 12259964326927110866866776228808161337250421224373748224, 21812926725659065797324660502998994022561529591086874194578215566049280

Offset: 1

Peter Kagey, Dec 16 2023

nonn

Dan Davis, On a Tiling Scheme from M. C. Escher, Electron. J. Combin. 4(2) (1996), #R23.
Peter Kagey, Illustration of a(2)=154
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-23.

Cf. A255016, A295223, A367525, A367533, A367534, A367535, A367536, A368137.

A368138[n_] := 1/(8n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 8^(n^2/LCM[c, d])]]]] + If[EvenQ[n], n^2 (3/4*8^(n^2/2) + 8^(n^2/4)) + n*DivisorSum[n, Function[c, EulerPhi[c] (If[EvenQ[c], 2*8^(n^2/c), 8^(n^2/(2 c))])]], 0] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 8^(n^2/(2 d)), 0]]])

A368139 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by two tiles that are each fixed under both diagonal and antidiagonal reflection.

Original entry on oeis.org

2, 6, 36, 1282, 340880, 477513804, 2872221202512, 72057600262282324, 7462505061854009276768, 3169126500572875969052992416, 5492677668532714149024993226980288, 38716571525226776302072008065489884436832, 1106936151351216411420647256070432280699273711360

Offset: 1

Peter Kagey, Dec 16 2023

nonn

S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015).
Peter Kagey, Illustration of a(3)=36
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-21, A-23.

Cf. A295223, A367526, A368140, A368141, A368142.

A368139[n_] := 1/(4n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + n^2*If[OddQ[n], 2^((n^2 + 1)/2), (7*2^(n^2/2 - 2))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*If[EvenQ[d], 2^(n^2/(2 d)), 2^((n^2 + n)/(2d))]]])

A368140 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by a tile that is fixed under only diagonal reflection.

Original entry on oeis.org

1, 4, 22, 1154, 337192, 477360876, 2872203226920, 72057597041056852, 7462505060326909791920, 3169126500571693774150807456, 5492677668532711895587506949961184, 38716571525226776294594927800946276718944, 1106936151351216411420589971585441310578379941760

Offset: 1

Peter Kagey, Dec 16 2023

nonn

Peter Kagey, Illustration of a(3)=22
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-21, A-24.

Cf. A295223, A367526, A368139, A368141, A368142.

A368140[n_] := 1/(4n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + n^2*If[EvenQ[n], (3*2^(n^2/2 - 2)), 0] + n*DivisorSum[n, Function[d, EulerPhi[d] If[EvenQ[d], 2^(n^2/(2 d) + 1), 2^((n^2 + n)/(2d))]]])

A367536 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under matrix transposition but no other symmetries.

Original entry on oeis.org

1, 17, 3692, 33572458, 5629501212064, 16397105857614447792, 808450637900676611412052288, 664613997892457939730524059906099232, 9021615045252487149405529092893182593313188608, 2008672555323737844427452615629277349189270615385935288832

Offset: 1

Peter Kagey, Dec 13 2023

nonn

A Truchet tile is an example of a tile that is fixed under matrix transposition but no other symmetries.

Peter Kagey, Illustration of a(2)=17
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-21, A-23.
Eric Weisstein's World of Mathematics, Truchet Tiling

Cf. A255016, A295223, A302484, A367533, A367534, A367535.

A367536[n_] := 1/(8n^2) (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] +If[OddQ[n],n*DivisorSum[n, Function[c, EulerPhi[c] 2^(n^2/c + 1)]],n*DivisorSum[n, Function[c, EulerPhi[c] (4^(n^2/LCM[2, c]) + 2^(n^2/c + 1) + If[OddQ[c], 0, 4^(n^2/c)])]] + n^2 (3*2^(n^2 - 2) + 2^(n^2/2))])

cp's OEIS Frontend

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

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A367533 The number of ways of tiling the n X n torus 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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A367534 The number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 90-degree rotation but not reflection.

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A367535 The number of ways of tiling the n X n torus 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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A368137 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under 180-degree rotation.

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A368138 Number of ways of tiling the n X n torus up to the symmetries of the square by an asymmetric tile.

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A368139 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by two tiles that are each fixed under both diagonal and antidiagonal reflection.

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A368140 Number of ways of tiling the n X n torus up to diagonal and antidiagonal reflection of the square by a tile that is fixed under only diagonal reflection.

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A367536 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under matrix transposition but no other symmetries.

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