A368260 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflections by two tiles that are each fixed under vertical reflection.

2, 3, 3, 6, 7, 4, 10, 24, 14, 6, 20, 76, 100, 40, 8, 36, 288, 700, 564, 108, 14, 72, 1072, 5560, 8296, 3384, 362, 20, 136, 4224, 43800, 131856, 104968, 22288, 1182, 36, 272, 16576, 350256, 2098720, 3358736, 1399176, 150972, 4150, 60

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  14   100     700     5560      43800
   4 |  6  40   564    8296   131856    2098720
   5 |  8 108  3384  104968  3358736  107377488
   6 | 14 362 22288 1399176 89505984 5726689312

Mathematica

A368260[n_, m_] := 1/(2 n) (DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + If[EvenQ[m], DivisorSum[n,  EulerPhi[#]*2^(n*m/LCM[#, 2]) &], DivisorSum[n,  EulerPhi[#]*2^((n*m - n)/LCM[#, 2])*2^(n/#) &]])

Crossrefs

Sequence in context: A187262 A117670 A368253 * A368262 A181695 A322291

Adjacent sequences: A368257 A368258 A368259 * A368261 A368262 A368263

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved