A255016 -id:A255016 - cp's OEIS frontend

A295223 Number of tilings of the n X n torus, using diagonal lines to connect the gridpoints.

1, 4, 18, 669, 170440, 238773358, 1436110601256, 36028800332480074, 3731252530927004638384, 1584563250286480205777197264, 2746338834266357074512496613490144, 19358285762613388151183577985346072926384, 553468075675608205710323628035216140349636855680

Offset: 1

Peter Kagey, Nov 17 2017

nonn

			For n = 3, the following four tilings are considered equivalent:
*---*->-+---+   +---+->-*---*   *---*->-+---+   +---+->-+---+
| / | \ | \ |   | / | / | \ |   | \ | / | / |   | / | \ | \ |
*---*---+---+   +---+---*---*   *---*---+---+   *---*---+---+
^ / | / | \ ^ = ^ / | \ | \ ^ = ^ \ | / | \ ^ = ^ \ | / | / ^
+---+---+---+   +---+---+---+   +---+---+---+   *---*---+---+
| \ | / | / |   | \ | \ | / |   | / | \ | \ |   | \ | / | \ |
+---+->-+---+   +---+->-+---+   +---+->-+---+   +---+->-+---+
The transformations are horizontal reflection, shifting to the right, and shifting down.

Peter Kagey, Table of n, a(n) for n = 1..57
Peter Kagey, The eighteen tilings of the 3 x 3 torus.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A255016, A295229.

a[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^2/LCM[2, c]) + If[OddQ[c], 0, 2^(n^2/c)])]], n*DivisorSum[n, Function[c, EulerPhi[c]*If[OddQ[c], 0, 2^(n^2/c)]]]] + If[OddQ[n], 0, n^2 (2^(n^2/4))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 2^((n^2 + n)/(2 d)), EvenQ[d], 2^(n^2/(2 d))]]])

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]]])

A255015 Number of toroidal n X n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition.

Original entry on oeis.org

1, 2, 6, 44, 2209, 674384, 954623404, 5744406453840, 144115192471496836, 14925010120653819583840, 6338253001142965335834871200, 10985355337065423791175013899922368, 77433143050453552587418968170813573149024

Offset: 0

Stewart N. Ethier, Feb 12 2015

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..57
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) # 15.8.3.
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.
Wikipedia, Polya enumeration theorem.

Cf. A184271 (number of m X n binary arrays allowing rotation of rows/columns), A179043 (main diagonal of A184271), A222188 (number of m X n binary arrays allowing rotation/reflection of rows/columns), A209251 (main diagonal of A222188), A255016 (number of n X n binary arrays allowing rotation/reflection of rows/columns as well as matrix transposition).

a[n_] := (2 n^2)^(-1) Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/ LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}] + (2 n)^(-1) Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n (n + d - 2 IntegerPart[d/2])/(2 d)), 0], {d, 1, n}];

a(n) = (2*n^2)^{-1} Sum_{ c divides n } Sum_{ d divides n } phi(c)*phi(d)* 2^(n^2/lcm(c,d)) + (2*n)^{-1} Sum_{ d divides n } phi(d)*2^(n*(n + d - 2 *floor(d/2))/(2*d)), where phi is A000010.

a(0)=1 from Alois P. Heinz, Feb 19 2015

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))])

A364794 Number of distinct binary arrays of size n X n with respect to isometric transformations.

Original entry on oeis.org

1, 2, 6, 86, 7626, 3956996, 8326366368, 69277957195904, 2287898999182608384, 301053169143557925650432, 158147142250171927345054089216, 331982638848895606930198405868158976, 2786232352655643085145552249123037486514176

Offset: 0

Johnny Sammon, Aug 08 2023

			For n = 2, the a(2) = 6 distinct binary arrays are
  OO  XO  XX  XO  XX  XX
  OO  OO  OO  OX  XO  XX
For n = 4
  OOXX                                OOXO
  OXXO  is considered equivalent to   XXXX
  OOXO                                OOOX
  OOXO                                OOOO
because we can rotate the bounding box of the Xs 90 degrees clockwise and place it back into the array as given above.

Johnny Sammon, Counting Distinct Binary Arrays With Respect To Isometric Transformations.

Cf. A054247 (without shift), A255016 (on n X n torus).

cp's OEIS Frontend

A295223 Number of tilings of the n X n torus, using diagonal lines to connect the gridpoints.

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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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A255015 Number of toroidal n X n binary arrays, allowing rotation of rows and/or columns as well as matrix transposition.

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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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A364794 Number of distinct binary arrays of size n X n with respect to isometric transformations.

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