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
Examples
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
Links
- 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.
Programs
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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)])