A368263 -id:A368263 - cp's OEIS frontend

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.

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])

A368264 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder by two distinct tiles.

Original entry on oeis.org

2, 4, 3, 8, 10, 4, 16, 36, 24, 6, 32, 136, 176, 70, 8, 64, 528, 1376, 1044, 208, 14, 128, 2080, 10944, 16456, 6560, 700, 20, 256, 8256, 87424, 262416, 209728, 43800, 2344, 36, 512, 32896, 699136, 4195360, 6710912, 2796976, 299600, 8230, 60

Offset: 1

Peter Kagey, Dec 21 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 |  4  24   176    1376     10944       87424
   4 |  6  70  1044   16456    262416     4195360
   5 |  8 208  6560  209728   6710912   214748416
   6 | 14 700 43800 2796976 178962784 11453291200

Peter Kagey, Illustration of T(2,3)=36
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A117401, A368257, A368259, A368261, A368263.

A368264[n_, m_] := 1/n (DivisorSum[n, EulerPhi[#]*2^(n*m/#) &])

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica