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%I #23 Nov 30 2023 12:26:17
%S 1,6,84,8548,4203520,8590557312,70368815480832,2305843028004192256,
%T 302231454912728264605696,158456325028538104598816096256,
%U 332306998946228986960926214931349504,2787593149816327892769293535238052808491008
%N Number of tilings of the n X n grid, using diagonal lines to connect the grid points.
%C The grids are counted up to reflection and rotation.
%C a(n) <= A295223(n).
%H Peter Kagey, <a href="/A295229/b295229.txt">Table of n, a(n) for n = 1..57</a>
%H Andrew Howroyd, <a href="/A295229/a295229.txt">Derivation of Formula</a>
%H Peter Kagey, <a href="/A295229/a295229.pdf">Example of the 6 tilings of the 2 X 2 grid</a>.
%H 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.
%F From _Andrew Howroyd_, Nov 19 2017: (Start)
%F a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 3*2^(n^2/2) + 2*2^(n^2/4)) / 8 for n even.
%F a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 2^((n^2+1)/2)) / 8 for n odd. (End)
%e For n = 2, the a(2) = 6 tilings are:
%e //, \/, /\, \\, /\, and \/.
%e // // // // \/ /\
%t Array[(2^(#^2) + 2*2^(# (# + 1)/2) + If[EvenQ@ #, 3*2^(#^2/2) + 2*2^(#^2/4), 2^((#^2 + 1)/2)])/8 &, 12] (* _Michael De Vlieger_, Apr 12 2018 *)
%o (PARI) a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + if(n%2, 2^((n^2+1)/2), 3*2^(n^2/2) + 2*2^(n^2/4)))/8; \\ _Andrew Howroyd_, Nov 19 2017
%Y Cf. A054247, A295223.
%K nonn
%O 1,2
%A _Peter Kagey_, Nov 18 2017
%E a(5)-a(12) from _Andrew Howroyd_, Nov 19 2017
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