A368145 Number of ways of tiling the n X n torus up to 90-degree rotations of the square by an asymmetric tile.

1, 23, 7296, 67124336, 11258999068672, 32794211700912314368, 1616901275801313012113145856, 1329227995784915876578744357489750016, 18043230090504974298810923860695296894480941056, 4017345110647475688854905231100098373350012499289786810368

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

Table of n, a(n) for n=1..10.

Peter Kagey, Illustration of a(2)=23

Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-21, A-25.

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

Cf. A367530, A367532, A368138, A368142, A368143, A368144.

Sequence in context: A193430 A233213 A368142 * A090674 A013728 A028693

Adjacent sequences: A368142 A368143 A368144 * A368146 A368147 A368148

Keyword

nonn

Author

Peter Kagey, Dec 16 2023

Status

approved