A368222 -id:A368222 - 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]])

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]])

A368221 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by two tiles that are each fixed under horizontal reflection.

Original entry on oeis.org

2, 4, 3, 8, 10, 6, 16, 36, 40, 10, 32, 136, 288, 136, 20, 64, 528, 2176, 2080, 544, 36, 128, 2080, 16896, 32896, 16640, 2080, 72, 256, 8256, 133120, 524800, 526336, 131328, 8320, 136, 512, 32896, 1056768, 8390656, 16793600, 8390656, 1050624, 32896, 272

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  2    4      8      16        32          64
   2 |  3   10     36     136       528        2080
   3 |  6   40    288    2176     16896      133120
   4 | 10  136   2080   32896    524800     8390656
   5 | 20  544  16640  526336  16793600   537001984
   6 | 36 2080 131328 8390656 536887296 34359869440

Peter Kagey, Illustration of T(3,2)=40
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A368218, A368222.

A368221[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n], 2^(n*m/2), 2^(m (n + 1)/2)])

A368306 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 8, 9, 4, 4, 24, 32, 26, 4, 8, 56, 186, 182, 62, 9, 10, 190, 1096, 2130, 1096, 205, 10, 20, 596, 7356, 26296, 26380, 7356, 623, 22, 30, 2102, 49940, 350316, 671104, 350584, 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         8
   2 | 2   5    8     24       56       190
   3 | 2   9   32    186     1096      7356
   4 | 4  26  182   2130    26296    350316
   5 | 4  62 1096  26380   671104  17899020
   6 | 9 205 7356 350584 17897924 954481360

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. A368222, A368259, A368304, A368305, A368308, A184271.

A368306[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*4^(m/#)*Boole[EvenQ[#]]) &]/2, DivisorSum[m, EulerPhi[#]*2^(n*m/#) &, EvenQ]])

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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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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A368221 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by two tiles that are each fixed under horizontal reflection.

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A368306 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.

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