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a(n) = (2^(n^2)+2^((n^2+n)/2))/2.
1

%I #19 Nov 30 2023 11:57:38

%S 1,2,12,288,33280,16793600,34360786944,281475110928384,

%T 9223372071214514176,1208925819632221360750592,

%U 633825300114132715146861084672,1329227995784915909797295207699447808,11150372599265311570918974863776009399828480

%N a(n) = (2^(n^2)+2^((n^2+n)/2))/2.

%C The number of inequivalent n x n black and white checkerboards where two boards are indistinguishable if and only if they are identical after being reflected about the main diagonal.

%C Cycle index = (s_1^(n^2) + s_1^n*s_2^((n^2-n)/2))/2.

%H Michael De Vlieger, <a href="/A200564/b200564.txt">Table of n, a(n) for n = 0..57</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.

%t Table[(2^(n^2)+2^((n^2+n)/2))/2,{n,0,10}]

%o (PARI) a(n)=2^(n^2-1) + 2^((n^2+n)/2-1) \\ _Charles R Greathouse IV_, Dec 08 2022

%K nonn,easy

%O 0,2

%A _Geoffrey Critzer_, Nov 19 2011