A302484 Number of Truchet tilings of an n X n square up to rotation and reflection.

1, 1, 43, 32896, 536911936, 140737496743936, 590295810401655390208, 39614081257132309534260330496, 42535295865117307944451040976113238016, 730750818665451459101843020821051317142553624576, 200867255532373784442745261543437606940418017880259520626688

Offset: 0

Author

David Radcliffe, Apr 08 2018

Keywords

• nonn
• easy

Comments

A Truchet tile is a unit square split along the diagonal into two triangles, one black and the other white. It has four orientations, with the white half at the NW, NE, SE, and SW. There are 4^(n^2) ways to tile an n X n square with Truchet tiles if rotations and reflections are counted as different. The number of tilings up to symmetry can be found using Burnside's lemma.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..40
• Peter Kagey, Illustration of the a(2) = 43 tilings of the 2 X 2 grid.
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.
• Eric Weisstein's World of Mathematics, Truchet Tiling
• Wikipedia, Truchet tiles

Programs

Mathematica

f[n_]:=If[EvenQ[n],(4^(n^2) + 5 4^(n^2/2) + 2 4^(n^2/4))/8, (4^(n^2) + 2 4^(n^2/2))/8]; Join[{1}, Array[f, 60]] (* Vincenzo Librandi, Apr 09 2018 *)

Python

def a(n): return (4**(n*n)+2**(n*n+1))//8 if n%2 else (4**(n*n)+5*4**(n*n//2)+2*4**(n*n//4))//8

Formula

a(n) = (4^(n^2) + 5*4^(n^2/2) + 2*4^(n^2/4)) / 8 if n is even.

a(n) = (4^(n^2) + 2*4^(n^2/2)) / 8 if n is odd.