A368259 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflection by an asymmetric tile.

1, 2, 2, 4, 6, 2, 8, 20, 12, 4, 16, 72, 88, 39, 4, 32, 272, 688, 538, 104, 9, 64, 1056, 5472, 8292, 3280, 366, 10, 128, 4160, 43712, 131464, 104864, 22028, 1172, 22, 256, 16512, 349568, 2098704, 3355456, 1399512, 149800, 4179, 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   2     4       8       16         32
   2 | 2   6    20      72      272       1056
   3 | 2  12    88     688     5472      43712
   4 | 4  39   538    8292   131464    2098704
   5 | 4 104  3280  104864  3355456  107374208
   6 | 9 366 22028 1399512 89489584 5726711136

Mathematica

A368259[n_, m_]:=1/(2n) (DivisorSum[n, EulerPhi[#]*2^(n*m/#)&]+n*2^(n*m/2-1)*Boole[EvenQ[n]])

Crossrefs

Sequence in context: A319579 A286631 A286243 * A305125 A260095 A209999

Adjacent sequences: A368256 A368257 A368258 * A368260 A368261 A368262

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved