A368256 -id:A368256 - cp's OEIS frontend

A368303 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotations but not horizontal or vertical reflections.

1, 2, 2, 2, 5, 2, 4, 8, 8, 4, 4, 22, 24, 22, 4, 8, 44, 120, 120, 44, 8, 9, 135, 612, 1203, 612, 135, 9, 18, 362, 3892, 13600, 13600, 3892, 362, 18, 23, 1211, 25482, 177342, 337600, 177342, 25482, 1211, 23, 44, 3914, 176654, 2404372, 8962618, 8962618, 2404372, 176654, 3914, 44

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     22      44       135
   3 | 2   8   24    120     612      3892
   4 | 4  22  120   1203   13600    177342
   5 | 4  44  612  13600  337600   8962618
   6 | 8 135 3892 177342 8962618 477371760

Peter Kagey, Illustration of T(3,3)=24
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A222188, A368219, A368256, A368302, A368304, A368307.

A368303[n_, m_]:=1/(4*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, Function[c, EulerPhi[c]EulerPhi[d]2^(m*n/LCM[c, d])]]]] + If[EvenQ[n], n/2*DivisorSum[m, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2m/#)*Boole[EvenQ[#]])&], n*DivisorSum[m, EulerPhi[#](2^(n*m/#))&, EvenQ]] + If[EvenQ[m], m/2*DivisorSum[n, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((m - 2)*n/LCM[2, #])*2^(2n/#)*Boole[EvenQ[#]])&], m*DivisorSum[n, EulerPhi[#](2^(m*n/#))&, EvenQ]] + n*m*2^((n*m)/2)*Which[OddQ[n*m], Sqrt[2], OddQ[n + m], 3/2, True, 7/4])

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])

cp's OEIS Frontend

A368303 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotations but not horizontal or vertical reflections.

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