A368220 -id:A368220 - cp's OEIS frontend

A368218 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under horizontal reflection only.

1, 3, 2, 4, 7, 3, 10, 20, 24, 6, 16, 76, 144, 76, 10, 36, 272, 1120, 1056, 288, 20, 64, 1072, 8448, 16576, 8320, 1072, 36, 136, 4160, 66816, 262656, 263680, 65792, 4224, 72, 256, 16576, 528384, 4197376, 8396800, 4197376, 525312, 16576, 136

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k |  1    2     3       4         5           6
  ----+--------------------------------------------
    1 |  1    3     4      10        16          36
    2 |  2    7    20      76       272        1072
    3 |  3   24   144    1120      8448       66816
    4 |  6   76  1056   16576    262656     4197376
    5 | 10  288  8320  263680   8396800   268517376
    6 | 20 1072 65792 4197376 268451840 17180065792

Peter Kagey, Illustration of T(3,2)=24
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-1, A-3.

Cf. A225910, A367524, A368219, A368220, A368221.

A368218[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + Boole[EvenQ[n*m]] + Boole[EvenQ[m]] + If[EvenQ[n], 1, 2^(m/2)])

A368219 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflection.

Original entry on oeis.org

1, 2, 2, 3, 7, 3, 6, 20, 20, 6, 10, 76, 136, 76, 10, 20, 272, 1056, 1056, 272, 20, 36, 1072, 8256, 16576, 8256, 1072, 36, 72, 4160, 65792, 262656, 262656, 65792, 4160, 72, 136, 16576, 524800, 4197376, 8390656, 4197376, 524800, 16576, 136

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k |  1    2     3       4         5           6
  ----+--------------------------------------------
    1 |  1    2     3       6        10          20
    2 |  2    7    20      76       272        1072
    3 |  3   20   136    1056      8256       65792
    4 |  6   76  1056   16576    262656     4197376
    5 | 10  272  8256  262656   8390656   268451840
    6 | 20 1072 65792 4197376 268451840 17180065792

Peter Kagey, Illustration of T(3,2)=20
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-1, A-3.

Cf. A225910, A368218, A368220, A368223.

A368219[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + If[EvenQ[n*m], 1, Sqrt[2]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])

A368222 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by an asymmetric tile.

Original entry on oeis.org

1, 2, 3, 4, 10, 4, 8, 36, 32, 10, 16, 136, 256, 136, 16, 32, 528, 2048, 2080, 512, 36, 64, 2080, 16384, 32896, 16384, 2080, 64, 128, 8256, 131072, 524800, 524288, 131328, 8192, 136, 256, 32896, 1048576, 8390656, 16777216, 8390656, 1048576, 32896, 256

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  1    2      4       8        16          32
   2 |  3   10     36     136       528        2080
   3 |  4   32    256    2048     16384      131072
   4 | 10  136   2080   32896    524800     8390656
   5 | 16  512  16384  524288  16777216   536870912
   6 | 36 2080 131328 8390656 536887296 34359869440

Peter Kagey, Illustration of T(3,2)=32
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A368220, A368221, A368224, A117401.

A368222[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n]])

A368224 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by an asymmetric tile.

Original entry on oeis.org

1, 3, 3, 4, 10, 4, 10, 36, 36, 10, 16, 136, 256, 136, 16, 36, 528, 2080, 2080, 528, 36, 64, 2080, 16384, 32896, 16384, 2080, 64, 136, 8256, 131328, 524800, 524800, 131328, 8256, 136, 256, 32896, 1048576, 8390656, 16777216, 8390656, 1048576, 32896, 256

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  1    3      4      10        16          36
   2 |  3   10     36     136       528        2080
   3 |  4   36    256    2080     16384      131328
   4 | 10  136   2080   32896    524800     8390656
   5 | 16  528  16384  524800  16777216   536887296
   6 | 36 2080 131328 8390656 536887296 34359869440

Peter Kagey, Illustration of T(3,3)=256
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A368220, A368222, A368223, A117401.

A368224[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n*m]])

A368304 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by an asymmetric tile.

Original entry on oeis.org

1, 4, 4, 6, 28, 6, 23, 194, 194, 23, 52, 2196, 7296, 2196, 52, 194, 26524, 350573, 350573, 26524, 194, 586, 351588, 17895736, 67136624, 17895736, 351588, 586, 2131, 4798174, 954495904, 13744131446, 13744131446, 954495904, 4798174, 2131

Offset: 1

Peter Kagey, Dec 21 2023

			Table begins:
  n\k|   1      2         3             4                5
  ---+----------------------------------------------------
   1 |   1      4         6            23               52
   2 |   4     28       194          2196            26524
   3 |   6    194      7296        350573         17895736
   4 |  23   2196    350573      67136624      13744131446
   5 |  52  26524  17895736   13744131446   11258999068672
   6 | 194 351588 954495904 2932037300956 9607679419823148

Peter Kagey, Illustration of T(2,2)=28
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A222188, A368220, A368257, A368302, A368303, A368306, A368308, A184271.

A368304[n_,m_]:=1/(4*n*m) (DivisorSum[n, Function[d,DivisorSum[m,Function[c,EulerPhi[c]EulerPhi[d]4^(m*n/LCM[c,d])]]]]+If[EvenQ[n],n/2*DivisorSum[m, EulerPhi[#](4^(n*m/LCM[2,#])+4^((n-2)*m/LCM[2,#])*4^(2m/#)*Boole[EvenQ[#]])&],n*DivisorSum[m,EulerPhi[#](4^(n*m/#))&,EvenQ]]+If[EvenQ[m], m/2*DivisorSum[n,EulerPhi[#](4^(n*m/LCM[2,#])+4^((m-2)*n/LCM[2,#])*4^(2n/#)*Boole[EvenQ[#]])&],m*DivisorSum[n, EulerPhi[#](4^(m*n/#))&,EvenQ]]+Which[EvenQ[n]&&EvenQ[m],(n*m)/4 (3*2^(n*m)),OddQ[n*m],0,OddQ[n+m],(n*m)/2 (2^(n*m))])

cp's OEIS Frontend

A368218 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under horizontal reflection only.

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A368219 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflection.

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A368222 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by an asymmetric tile.

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A368224 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by an asymmetric tile.

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A368304 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by an asymmetric tile.

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