A368253 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by two tiles that are fixed under these reflections.

2, 3, 3, 6, 7, 4, 10, 24, 13, 6, 20, 76, 74, 34, 8, 36, 288, 430, 378, 78, 13, 72, 1072, 3100, 4756, 1884, 237, 18, 136, 4224, 23052, 70536, 53764, 11912, 687, 30, 272, 16576, 179736, 1083664, 1689608, 709316, 77022, 2299, 46

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  13    74    430     3100      23052
    4 |  6  34   378   4756    70536    1083664
    5 |  8  78  1884  53764  1689608   53762472
    6 | 13 237 11912 709316 44900448 2865540112

Mathematica

A368253[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*If[EvenQ[n], 1/2 (2^((n*m + 2 m)/2) + 2^(n*m/2)), 2^((n*m + m)/2)] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^((n*m - n)/LCM[d, 2])*2^(n/d)]]] + n*Which[EvenQ[m], 2^(n*m/2), OddQ[m] && EvenQ[n], (3/2*2^(n*m/2)), OddQ[m] && OddQ[n], 2^((n*m + 1)/2)])

Crossrefs

Cf. A222188, A225910.

Cf. A005418 (n=1), A225826 (n=2), A000029 (k=1), A222187 (k=2).

Sequence in context: A187763 A187262 A117670 * A368260 A368262 A181695

Adjacent sequences: A368250 A368251 A368252 * A368254 A368255 A368256

Keyword

nonn,tabl

Author

Peter Kagey, Dec 19 2023

Status

approved