A368262 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 two tiles that are each fixed under 180-degree rotation.

2, 3, 3, 6, 7, 4, 10, 24, 16, 6, 20, 76, 104, 43, 8, 36, 288, 720, 570, 120, 13, 72, 1072, 5600, 8356, 3408, 382, 18, 136, 4224, 43968, 131976, 105376, 22284, 1236, 30, 272, 16576, 350592, 2099728, 3359552, 1400536, 150824, 4243, 46

Offset

1,1

Links

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

Peter Kagey, Illustration of T(2,3)=24

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 |  2   3     6      10       20         36
   2 |  3   7    24      76      288       1072
   3 |  4  16   104     720     5600      43968
   4 |  6  43   570    8356   131976    2099728
   5 |  8 120  3408  105376  3359552  107390592
   6 | 13 382 22284 1400536 89505968 5726776672

Mathematica

A368262[n_, m_] := 1/(2n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 3/2, True, Sqrt[2]])

Crossrefs

Sequence in context: A117670 A368253 A368260 * A181695 A322291 A184830

Adjacent sequences: A368259 A368260 A368261 * A368263 A368264 A368265

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved