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.

0, 2, 4, 19, 80, 638, 6054, 76692, 1137284, 19405244, 370597430, 7825459362, 180862277352, 4540781512946, 123053646087312, 3580073396748560, 111297799861936256, 3682093529146577694, 129163727524848878358, 4788738149626920381804, 187102616692953377567060

Offset: 1

Author

Marko Riedel, Jun 29 2017

Keywords

• nonn

Comments

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.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Marko Riedel, Hexagonal tiles
• Marko Riedel, Maple code to compute number of tiles by ordinary enumeration and by Power Group Enumeration
• Marko Riedel, Maple code for number of tiles using Burnside lemma.

Crossrefs

See A053871 for tiles with no symmetries being taken into account, A289191 for tiles with rotational symmetries only being taken into account.

Programs

PARI

\\ here R(n) is A289191.
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))}
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}
a(n)={(R(n) + S(n))/2} \\ Andrew Howroyd, Jan 26 2020

Extensions

Terms a(14) and beyond from Andrew Howroyd, Jan 26 2020