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A368145 The number of ways of tiling the n X n torus up to 90-degree rotations of the square by an asymmetric tile. 3
1, 23, 7296, 67124336, 11258999068672, 32794211700912314368, 1616901275801313012113145856, 1329227995784915876578744357489750016, 18043230090504974298810923860695296894480941056, 4017345110647475688854905231100098373350012499289786810368 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
M.C. Escher enumerated a(2) = 23 by hand in May 1942, being perhaps the first person to attempt this sort of counting problem. (See Doris Schattschneider's book in the references for more details.)
REFERENCES
Doris Schattschneider, Visions of Symmetry, W.H. Freeman, 1990, pages 44-48.
LINKS
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.
Doris Schattschneider, Escher's combinatorial patterns, Electron. J. Combin. 4(2) (1996), #R17.
MATHEMATICA
A368145[n_] := 1/(4n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] + n^2*If[OddQ[n], 0, 3/4*2^n^2 + 2^(n^2/2)])
CROSSREFS
Sequence in context: A193430 A233213 A368142 * A090674 A013728 A028693
KEYWORD
nonn
AUTHOR
Peter Kagey, Dec 16 2023
STATUS
approved

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Last modified June 25 17:15 EDT 2024. Contains 373706 sequences. (Running on oeis4.)