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, 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.
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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, 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.
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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
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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
- 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.
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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]]])
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, 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
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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))])
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