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.

0, 2, 4, 22, 112, 1060, 11292, 149448, 2257288, 38720728, 740754220, 15648468804, 361711410384, 9081485302372, 246106843197984, 7160143986526240, 222595582448849152, 7364186944683168828, 258327454310582805036, 9577476294162996275928, 374205233351106756670120

Offset: 1

Author

Marko Riedel, Jun 27 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.

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

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 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.

Cf. A132102.

Programs

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

Extensions

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