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.

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

1,2

Links

Table of n, a(n) for n=1..45.

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.

Example

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

Mathematica

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]])

Crossrefs

Cf. A225910, A368218, A368220, A368223.

Sequence in context: A363365 A208151 A203362 * A183407 A222838 A222944

Adjacent sequences: A368216 A368217 A368218 * A368220 A368221 A368222

Keyword

nonn,tabl

Author

Peter Kagey, Dec 18 2023

Status

approved