A368257 -id:A368257 - cp's OEIS frontend

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.

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))])

A368255 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by a tile that is fixed under vertical reflections but not horizontal reflections.

Original entry on oeis.org

1, 2, 2, 3, 5, 2, 6, 14, 9, 4, 10, 44, 50, 26, 4, 20, 152, 366, 298, 62, 9, 36, 560, 2780, 4244, 1692, 205, 10, 72, 2144, 22028, 66184, 52740, 11272, 623, 22, 136, 8384, 175128, 1050896, 1679368, 701124, 75486, 2171, 30

Offset: 1

Peter Kagey, Dec 21 2023

			Table begins:
  n\k|  1   2     3       4          5            6
  ---+---------------------------------------------
   1 |  1   2     3       6         10           20
   2 |  2   5    14      44        152          560
   3 |  2   9    50     366       2780        22028
   4 |  4  26   298    4244      66184      1050896
   5 |  4  62  1692   52740    1679368     53696936
   6 |  9 205 11272  701124   44761184   2863442960
   7 | 10 623 75486 9591666 1227208420 157073688884

Peter Kagey, Illustration of T(2,3)=14
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A368218, A368253, A368254, A368256, A368257, A368260.

A368255[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*(2^(n*m/2 - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^((n*m - n)/LCM[d, 2])*2^(n/d)]]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])

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

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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A368255 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by a tile that is fixed under vertical reflections but not horizontal reflections.

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Mathematica

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

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica