A368257 Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by an asymmetric tile.

1, 6, 4, 16, 44, 6, 72, 544, 366, 23, 256, 8384, 21856, 4244, 52, 1056, 131584, 1399512, 1050128, 52740, 194, 4096, 2100224, 89478656, 268472384, 53687104, 701124, 586, 16512, 33562624, 5726711136, 68719870208, 54975896016, 2863399264, 9591666, 2131

Offset

1,2

Links

Table of n, a(n) for n=1..36.

Peter Kagey, Illustration of T(2,2)=44

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
  ---+-------------------------------------------------------
   1 |   1      6         16             72               256
   2 |   4     44        544           8384            131584
   3 |   6    366      21856        1399512          89478656
   4 |  23   4244    1050128      268472384       68719870208
   5 |  52  52740   53687104    54975896016    56294995342336
   6 | 194 701124 2863399264 11728132423744 48038396383286784

Mathematica

A368257[n_, m_] := 1/(4n)*(DivisorSum[n, EulerPhi[#]*4^(n*m/#) &] + n (2^(n*m - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*4^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*4^(n*m/#) &, EvenQ]] + n*2^(n*m)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])

Crossrefs

Sequence in context: A198459 A083581 A171089 * A180495 A213761 A160248

Adjacent sequences: A368254 A368255 A368256 * A368258 A368259 A368260

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved