A368261 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflection by an asymmetric tile.

1, 3, 2, 4, 7, 2, 10, 20, 14, 4, 16, 76, 88, 40, 4, 36, 272, 700, 532, 108, 8, 64, 1072, 5472, 8296, 3280, 362, 10, 136, 4160, 43800, 131344, 104968, 21944, 1182, 20, 256, 16576, 349568, 2098720, 3355456, 1399176, 149800, 4150, 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   3     4      10       16         36
   2 | 2   7    20      76      272       1072
   3 | 2  14    88     700     5472      43800
   4 | 4  40   532    8296   131344    2098720
   5 | 4 108  3280  104968  3355456  107377488
   6 | 8 362 21944 1399176 89484128 5726689312

Mathematica

A368261[n_, m_]:=1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#)&] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*2^(n*m/LCM[#, 2])&], DivisorSum[n, EulerPhi[#]*2^(n*m/#)&, EvenQ]])

Crossrefs

Sequence in context: A317736 A060006 A368254 * A368263 A123097 A368218

Adjacent sequences: A368258 A368259 A368260 * A368262 A368263 A368264

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved