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A289269 Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational and reflectional, i.e. dihedral, symmetry.

%I A289269 #27 Jan 26 2020 20:57:55
%S A289269 0,2,4,19,80,638,6054,76692,1137284,19405244,370597430,7825459362,
%T A289269 180862277352,4540781512946,123053646087312,3580073396748560,
%U A289269 111297799861936256,3682093529146577694,129163727524848878358,4788738149626920381804,187102616692953377567060
%N A289269 Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational and reflectional, i.e. dihedral, symmetry.
%C A289269 The case n=2 is a degenerate polygon (two sides connecting two vertices). The two possibilities are when the edges cross and do not cross. Polygons start at n=3 with a triangle.
%H A289269 Andrew Howroyd, <a href="/A289269/b289269.txt">Table of n, a(n) for n = 1..200</a>
%H A289269 Marko Riedel, <a href="https://math.stackexchange.com/questions/2097450/">Hexagonal tiles</a>
%H A289269 Marko Riedel, <a href="/A289269/a289269.maple.txt">Maple code to compute number of tiles by ordinary enumeration and by Power Group Enumeration</a>
%H A289269 Marko Riedel, <a href="/A289269/a289269_1.maple.txt">Maple code for number of tiles using Burnside lemma.</a>
%o A289269 (PARI) \\ here R(n) is A289191.
%o A289269 S(n)={sum(i=0, n\2, (-1)^i * sum(j=0, (n-2*i)\2, (2*j)!/j! * if(n%2, if(j, 2*binomial(n\2, i)*binomial(n-2*i-1, 2*j-1)), binomial(n/2, i)*binomial(n-2*i, 2*j) + if(j, binomial(n/2-1, i)*binomial(n-2*i-2, 2*j-2))) / 2))}
%o A289269 R(n)={sumdiv(n, d, my(m=n/d); eulerphi(d)*sum(i=0, m, (-1)^i * binomial(m, i) * sum(j=0, m-i, (d%2==0 || m-i-j==0) * binomial(2*(m-i), 2*j) * d^j * (2*j)! / (j!*2^j) )))/n}
%o A289269 a(n)={(R(n) + S(n))/2} \\ _Andrew Howroyd_, Jan 26 2020
%Y A289269 See A053871 for tiles with no symmetries being taken into account, A289191 for tiles with rotational symmetries only being taken into account.
%K A289269 nonn
%O A289269 1,2
%A A289269 _Marko Riedel_, Jun 29 2017
%E A289269 Terms a(14) and beyond from _Andrew Howroyd_, Jan 26 2020