A368258 Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflections by two tiles that are each fixed under horizontal reflection.

2, 4, 3, 8, 10, 4, 16, 36, 20, 6, 32, 136, 120, 55, 8, 64, 528, 816, 666, 136, 13, 128, 2080, 5984, 9316, 3536, 430, 18, 256, 8256, 45760, 139656, 106912, 23052, 1300, 30, 512, 32896, 357760, 2164240, 3371840, 1415896, 151848, 4435, 46

Offset

1,1

Links

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

Peter Kagey, Illustration of T(2,3)=36

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    4      8       16         32           64
   2 |  3   10     36      136        528         2080
   3 |  4   20    120      816       5984        45760
   4 |  6   55    666     9316     139656      2164240
   5 |  8  136   3536   106912    3371840    107505280
   6 | 13  430  23052  1415896   89751728   5730905440
   7 | 18 1300 151848 19206736 2454791328 314154568000

Mathematica

A368258[n_, m_] := 1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#)&] + n*2^(n*m/2)*If[EvenQ[n], 1/2*(2^m + 1), 2^(m/2)])

Crossrefs

Sequence in context: A241909 A245451 A371183 * A368264 A026166 A186003

Adjacent sequences: A368255 A368256 A368257 * A368259 A368260 A368261

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved