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A289191 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 symmetry.

%I A289191 #31 Jan 26 2020 18:18:05
%S A289191 0,2,4,22,112,1060,11292,149448,2257288,38720728,740754220,
%T A289191 15648468804,361711410384,9081485302372,246106843197984,
%U A289191 7160143986526240,222595582448849152,7364186944683168828,258327454310582805036,9577476294162996275928,374205233351106756670120
%N A289191 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 symmetry.
%C A289191 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.
%C A289191 The sequence A132102 enumerates the case that edges are allowed between exits on the same side. This sequence can be enumerated in a similar manner using inclusion-exclusion on the number of sides that have their two exits connected. - _Andrew Howroyd_, Jan 26 2020
%H A289191 Andrew Howroyd, <a href="/A289191/b289191.txt">Table of n, a(n) for n = 1..200</a>
%H A289191 Marko Riedel, <a href="https://math.stackexchange.com/questions/2097450/">Hexagonal tiles</a>.
%H A289191 Marko Riedel, <a href="/A289191/a289191.maple.txt">Maple code to compute number of tiles by ordinary enumeration and by Power Group Enumeration.</a>
%H A289191 Marko Riedel, <a href="/A289191/a289191_1.maple.txt">Maple code for number of tiles using Burnside lemma.</a>
%o A289191 (PARI) a(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} \\ _Andrew Howroyd_, Jan 26 2020
%Y A289191 See A053871 for tiles with no rotational symmetries being taken into account, A289269 for tiles with rotational and reflectional symmetries being taken into account, A289343 for the same statistic evaluated when n is prime.
%Y A289191 Cf. A132102.
%K A289191 nonn
%O A289191 1,2
%A A289191 _Marko Riedel_, Jun 27 2017
%E A289191 Terms a(14) and beyond from _Andrew Howroyd_, Jan 26 2020