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A368308
Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.
4
1, 2, 2, 2, 5, 2, 4, 9, 9, 4, 4, 26, 32, 26, 4, 9, 62, 192, 192, 62, 9, 10, 205, 1096, 2174, 1096, 205, 10, 22, 623, 7440, 26500, 26500, 7440, 623, 22, 30, 2171, 49940, 351336, 671104, 351336, 49940, 2171, 30
OFFSET
1,2
LINKS
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 2 4 4 9
2 | 2 5 9 26 62 205
3 | 2 9 32 192 1096 7440
4 | 4 26 192 2174 26500 351336
5 | 4 62 1096 26500 671104 17904476
6 | 9 205 7440 351336 17904476 954546880
MATHEMATICA
A368308[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*m*2^(n*m/2)*Which[OddQ[n*m], 0, OddQ[n + m], 1/2, True, 3/4])
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Dec 21 2023
STATUS
approved