A368218 -id:A368218 - 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]])

A368219 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 a tile that is fixed under 180-degree rotation but not horizontal or vertical reflection.

Original entry on oeis.org

1, 2, 2, 3, 7, 3, 6, 20, 20, 6, 10, 76, 136, 76, 10, 20, 272, 1056, 1056, 272, 20, 36, 1072, 8256, 16576, 8256, 1072, 36, 72, 4160, 65792, 262656, 262656, 65792, 4160, 72, 136, 16576, 524800, 4197376, 8390656, 4197376, 524800, 16576, 136

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k |  1    2     3       4         5           6
  ----+--------------------------------------------
    1 |  1    2     3       6        10          20
    2 |  2    7    20      76       272        1072
    3 |  3   20   136    1056      8256       65792
    4 |  6   76  1056   16576    262656     4197376
    5 | 10  272  8256  262656   8390656   268451840
    6 | 20 1072 65792 4197376 268451840 17180065792

Peter Kagey, Illustration of T(3,2)=20
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. A225910, A368218, A368220, A368223.

A368219[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + If[EvenQ[n*m], 1, Sqrt[2]] + Boole[EvenQ[n]] + Boole[EvenQ[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)])

A368302 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 horizontal reflections but not vertical reflections.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 9, 8, 4, 4, 26, 22, 22, 4, 9, 62, 120, 126, 44, 8, 10, 205, 600, 1267, 592, 135, 9, 22, 623, 3936, 14164, 13600, 3936, 362, 18, 30, 2171, 25556, 181782, 337192, 178366, 25314, 1211, 23, 62, 7429, 177678, 2437726, 8965354, 8980642, 2404372, 176998, 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         9
   2 | 2   5    9     26      62       205
   3 | 2   8   22    120     600      3936
   4 | 4  22  126   1267   14164    181782
   5 | 4  44  592  13600  337192   8965354
   6 | 8 135 3936 178366 8980642 477655760

Peter Kagey, Illustration of T(3,3)=22
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A222188, A368218, A368254, A368303, A368304, A368305.

A368302[n_, m_] := 1/(4*n*m) (DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], 1/2*DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2 m/#)) &], DivisorSum[m, EulerPhi[#] (2^((n - 1)*m/LCM[2, #])*2^(m/#)) &]] + m*If[EvenQ[m], 1/2*DivisorSum[n, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^(m*n/#)*Boole[EvenQ[#]]) &], DivisorSum[n, EulerPhi[#]*2^(m*n/#) &, EvenQ]] + n*m*2^(n*m/2)*Which[EvenQ[n] && EvenQ[m], 3/4, OddQ[n*m], 0, OddQ[n + m], 1/2])

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.

Original entry on oeis.org

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

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A368219 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 a tile that is fixed under 180-degree rotation but not horizontal or vertical reflection.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A368302 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 horizontal reflections but not vertical reflections.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica