A368253 -id:A368253 - cp's OEIS frontend

A368254 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 horizontal reflections but not vertical reflections.

1, 3, 2, 4, 7, 2, 10, 20, 13, 4, 16, 76, 60, 34, 4, 36, 272, 430, 346, 78, 8, 64, 1072, 2992, 4756, 1768, 237, 9, 136, 4160, 23052, 70024, 53764, 11612, 687, 18, 256, 16576, 178880, 1083664, 1685920, 709316, 75924, 2299, 23

Offset: 1

Peter Kagey, Dec 19 2023

			Table begins:
  n\k| 1   2     3      4        5          6
  ---+---------------------------------------
   1 | 1   3     4     10       16         36
   2 | 2   7    20     76      272       1072
   3 | 2  13    60    430     2992      23052
   4 | 4  34   346   4756    70024    1083664
   5 | 4  78  1768  53764  1685920   53762472
   6 | 8 237 11612 709316 44881328 2865540112

Peter Kagey, Illustration of T(2,3)=20
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[n_, m_] := 1/(4n)(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*2^(n*m/2)*If[EvenQ[n], 1/2 (2^m + 1), 2^(m/2)] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)], EvenQ]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])

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

A368254 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 horizontal reflections but not vertical reflections.

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