A295223 Number of tilings of the n X n torus, using diagonal lines to connect the gridpoints.

1, 4, 18, 669, 170440, 238773358, 1436110601256, 36028800332480074, 3731252530927004638384, 1584563250286480205777197264, 2746338834266357074512496613490144, 19358285762613388151183577985346072926384, 553468075675608205710323628035216140349636855680

Offset

1,2

Links

Peter Kagey, Table of n, a(n) for n = 1..57

Peter Kagey, The eighteen tilings of the 3 x 3 torus.

Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Example

For n = 3, the following four tilings are considered equivalent:
*---*->-+---+   +---+->-*---*   *---*->-+---+   +---+->-+---+
| / | \ | \ |   | / | / | \ |   | \ | / | / |   | / | \ | \ |
*---*---+---+   +---+---*---*   *---*---+---+   *---*---+---+
^ / | / | \ ^ = ^ / | \ | \ ^ = ^ \ | / | \ ^ = ^ \ | / | / ^
+---+---+---+   +---+---+---+   +---+---+---+   *---*---+---+
| \ | / | / |   | \ | \ | / |   | / | \ | \ |   | \ | / | \ |
+---+->-+---+   +---+->-+---+   +---+->-+---+   +---+->-+---+
The transformations are horizontal reflection, shifting to the right, and shifting down.

Mathematica

a[n_] := 1/(8*n^2)*(DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d])]]]] + If[OddQ[n], n^2*2^((n^2 + 1)/2), n^2/4 (3*2^(n^2/2) + 2^((n^2 + 4)/2))] + 2*If[EvenQ[n], n/2*DivisorSum[n, Function[c, EulerPhi[c] (2^(n^2/LCM[2, c]) + If[OddQ[c], 0, 2^(n^2/c)])]], n*DivisorSum[n, Function[c, EulerPhi[c]*If[OddQ[c], 0, 2^(n^2/c)]]]] + If[OddQ[n], 0, n^2 (2^(n^2/4))] + 2*n*DivisorSum[n, Function[d, EulerPhi[d]*Which[OddQ[d], 2^((n^2 + n)/(2 d)), EvenQ[d], 2^(n^2/(2 d))]]])

Crossrefs

Cf. A255016, A295229.

Sequence in context: A082022 A354303 A114574 * A173222 A342467 A367533

Adjacent sequences: A295220 A295221 A295222 * A295224 A295225 A295226

Keyword

nonn

Author

Peter Kagey, Nov 17 2017

Status

approved