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, 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.
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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
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, 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.
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Table[{4096^(m^2 - m), 8^(m^2 - 1) (512^m^2 + 3*8^m^2 + 2)}, {m, 1, 5}] // Flatten
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))]]])
A367522
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 both horizontal and vertical reflection, but not diagonal reflection.
Original entry on oeis.org
1, 4, 84, 8292, 4203520, 8590033024, 70368815480832, 2305843010824323072, 302231454912728264605696, 158456325028529097399561355264, 332306998946228986960926214931349504, 2787593149816327892693735671512138485071872, 93536104789177786765036453099565034406633831137280
Offset: 1
- Michael De Vlieger, Table of n, a(n) for n = 1..57
- Peter Kagey, Illustration of a(3)=84.
- 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-6.
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a[n_] := If[EvenQ[n], 2^(#^2 - 3)*(2 + 3*2^#^2 + 8^#^2) &[n/2], 4^(#^2 - 2 # - 1)*(4^# + 4^#^2 + 8^#) &[(n + 1)/2]]; Array[a, 13] (* Michael De Vlieger, Jul 06 2024 *)
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