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

1,2

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.

Prog

(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

Crossrefs

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

Sequence in context: A064228 A226891 A362350 * A363303 A272988 A168246

Adjacent sequences: A289266 A289267 A289268 * A289270 A289271 A289272

Keyword

nonn

Author

Marko Riedel, Jun 29 2017

Extensions

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

Status

approved