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A302484 Number of Truchet tilings of an n X n square up to rotation and reflection. 7
1, 1, 43, 32896, 536911936, 140737496743936, 590295810401655390208, 39614081257132309534260330496, 42535295865117307944451040976113238016, 730750818665451459101843020821051317142553624576, 200867255532373784442745261543437606940418017880259520626688 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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 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
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.
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 *)
PROG
(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
CROSSREFS
Sequence in context: A356203 A184144 A262856 * A177488 A262648 A185558
KEYWORD
nonn,easy
AUTHOR
David Radcliffe, Apr 08 2018
STATUS
approved

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Last modified March 19 06:25 EDT 2024. Contains 370953 sequences. (Running on oeis4.)