A368224 -id:A368224 - cp's OEIS frontend

A368220 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 an asymmetric tile.

1, 6, 6, 16, 76, 16, 72, 1056, 1056, 72, 256, 16576, 65536, 16576, 256, 1056, 262656, 4196352, 4196352, 262656, 1056, 4096, 4197376, 268435456, 1073790976, 268435456, 4197376, 4096, 16512, 67117056, 17180000256, 274878431232, 274878431232, 17180000256, 67117056, 16512

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k |    1       2           3              4                  5
  ----+-----------------------------------------------------------
    1 |    1       6          16             72                256
    2 |    6      76        1056          16576             262656
    3 |   16    1056       65536        4196352          268435456
    4 |   72   16576     4196352     1073790976       274878431232
    5 |  256  262656   268435456   274878431232    281474976710656
    6 | 1056 4197376 17180000256 70368756760576 288230376688582656

Peter Kagey, Illustration of T(2,2)=76
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. A117401, A225910, A368218, A368219, A368222, A368224.

A368220[n_, m_] := 2^(n*m - 2)*(2^(n*m) + Boole[EvenQ[n*m]] + 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]])

A368223 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by two tiles that are each fixed under 180-degree rotation.

Original entry on oeis.org

2, 3, 3, 6, 10, 6, 10, 36, 36, 10, 20, 136, 272, 136, 20, 36, 528, 2080, 2080, 528, 36, 72, 2080, 16512, 32896, 16512, 2080, 72, 136, 8256, 131328, 524800, 524800, 131328, 8256, 136, 272, 32896, 1049600, 8390656, 16781312, 8390656, 1049600, 32896, 272

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  2    3      6      10        20          36
   2 |  3   10     36     136       528        2080
   3 |  6   36    272    2080     16512      131328
   4 | 10  136   2080   32896    524800     8390656
   5 | 20  528  16512  524800  16781312   536887296
   6 | 36 2080 131328 8390656 536887296 34359869440

Peter Kagey, Illustration of T(3,3)=272
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A368219, A368224.

A368223[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n*m], 2^(n*m/2), 2^((n*m + 1)/2)])

A368308 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 9, 9, 4, 4, 26, 32, 26, 4, 9, 62, 192, 192, 62, 9, 10, 205, 1096, 2174, 1096, 205, 10, 22, 623, 7440, 26500, 26500, 7440, 623, 22, 30, 2171, 49940, 351336, 671104, 351336, 49940, 2171, 30

Offset: 1

Peter Kagey, Dec 21 2023

			Table begins:
  n\k| 1   2    3      4        5         6
  ---+-------------------------------------
   1 | 1   2    2      4        4         9
   2 | 2   5    9     26       62       205
   3 | 2   9   32    192     1096      7440
   4 | 4  26  192   2174    26500    351336
   5 | 4  62 1096  26500   671104  17904476
   6 | 9 205 7440 351336 17904476 954546880

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

Cf. A184271, A368224, A368263, A368304, A368306, A368307.

A368308[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*m*2^(n*m/2)*Which[OddQ[n*m], 0, OddQ[n + m], 1/2, True, 3/4])

cp's OEIS Frontend

A368220 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 an asymmetric tile.

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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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A368223 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by two tiles that are each fixed under 180-degree rotation.

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A368308 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.

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