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

%I A302484 #27 Feb 16 2025 08:33:53
%S A302484 1,1,43,32896,536911936,140737496743936,590295810401655390208,
%T A302484 39614081257132309534260330496,42535295865117307944451040976113238016,
%U A302484 730750818665451459101843020821051317142553624576,200867255532373784442745261543437606940418017880259520626688
%N A302484 Number of Truchet tilings of an n X n square up to rotation and reflection.
%C A302484 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.
%H A302484 Michael De Vlieger, <a href="/A302484/b302484.txt">Table of n, a(n) for n = 0..40</a>
%H A302484 Peter Kagey, <a href="/A302484/a302484.pdf">Illustration of the a(2) = 43 tilings of the 2 X 2 grid</a>.
%H A302484 Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv:2311.13072 [math.CO], 2023. See p. 3.
%H A302484 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TruchetTiling.html">Truchet Tiling</a>
%H A302484 Wikipedia, <a href="http://en.wikipedia.org/wiki/Truchet_tiles">Truchet tiles</a>
%F A302484 a(n) = (4^(n^2) + 5*4^(n^2/2) + 2*4^(n^2/4)) / 8 if n is even.
%F A302484 a(n) = (4^(n^2) + 2*4^(n^2/2)) / 8 if n is odd.
%t A302484 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 *)
%o A302484 (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
%K A302484 nonn,easy
%O A302484 0,3
%A A302484 _David Radcliffe_, Apr 08 2018