A367536 Number of ways of tiling the n X n torus up to the symmetries of the square by a tile that is fixed under matrix transposition but no other symmetries.

1, 17, 3692, 33572458, 5629501212064, 16397105857614447792, 808450637900676611412052288, 664613997892457939730524059906099232, 9021615045252487149405529092893182593313188608, 2008672555323737844427452615629277349189270615385935288832

Offset

1,2

Comments

A Truchet tile is an example of a tile that is fixed under matrix transposition but no other symmetries.

Links

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

Peter Kagey, Illustration of a(2)=17

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-23.

Eric Weisstein's World of Mathematics, Truchet Tiling

Mathematica

A367536[n_] := 1/(8n^2) (DivisorSum[n, Function[d, DivisorSum[n, Function[c, EulerPhi[c] EulerPhi[d] 4^(n^2/LCM[c, d])]]]] +If[OddQ[n], n*DivisorSum[n, Function[c, EulerPhi[c] 2^(n^2/c + 1)]], n*DivisorSum[n, Function[c, EulerPhi[c] (4^(n^2/LCM[2, c]) + 2^(n^2/c + 1) + If[OddQ[c], 0, 4^(n^2/c)])]] + n^2 (3*2^(n^2 - 2) + 2^(n^2/2))])

Crossrefs

Cf. A255016, A295223, A302484, A367533, A367534, A367535.

Sequence in context: A125000 A228195 A032909 * A239165 A329168 A194015

Adjacent sequences: A367533 A367534 A367535 * A367537 A367538 A367539

Keyword

nonn

Author

Peter Kagey, Dec 13 2023

Status

approved