A368263 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to 180-degree rotation by an asymmetric tile.

1, 3, 2, 4, 7, 2, 10, 20, 16, 4, 16, 76, 88, 43, 4, 36, 272, 720, 538, 120, 9, 64, 1072, 5472, 8356, 3280, 382, 10, 136, 4160, 43968, 131464, 105376, 22028, 1236, 22, 256, 16576, 349568, 2099728, 3355456, 1400536, 149800, 4243, 30

Offset

1,2

Links

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

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.

Example

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  16    88     720     5472      43968
   4 | 4  43   538    8356   131464    2099728
   5 | 4 120  3280  105376  3355456  107390592
   6 | 9 382 22028 1400536 89489584 5726776672

Mathematica

A368263[n_, m_] := 1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])

Crossrefs

Sequence in context: A060006 A368254 A368261 * A123097 A368218 A352419

Adjacent sequences: A368260 A368261 A368262 * A368264 A368265 A368266

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved