A316198 Number of self-avoiding polygons with perimeter 2*n and sides = 1 that have vertex angles from the set 0, +-Pi/4, +-Pi/2, +-3*Pi/4, not counting rotations and reflections as distinct.

0, 2, 6, 59, 695, 12198, 240549, 5191160, 118346760, 2816763296

Offset

1,2

Comments

Holes are excluded, i.e., the boundary path may nowhere touch or intersect itself.

From Anton Pirogov, Jun 21 2026: (Start)

Also number of free simple matchstick polygons with perimeter 2*n on the cyclotomic ring Z[zeta_8].

The values a(7)-a(10) were determined by enumeration with the tilezz library using exact cyclotomic integer arithmetic over Z[zeta_8]. (End)

Links

Table of n, a(n) for n=1..10.

Contest Organizers, Al Zimmermann's Programming Contests - Snakes on a Plane, Fall 2006.

Hugo Pfoertner, Illustration of polygons of perimeter <= 10.

Anton Pirogov, tilezz - exact cyclotomic geometry library including a polygon enumerator.

Anton Pirogov, Rat Explorer - interactive database of cyclotomic matchstick polygons.

Anton Pirogov, RatDB datasets - reproducible datasets containing the enumerated polygons.

Crossrefs

Cf. A306178, A316195, A316197.

Sequence in context: A366366 A357092 A132525 * A074167 A083787 A347979

Adjacent sequences: A316195 A316196 A316197 * A316199 A316200 A316201

Keyword

nonn,walk,more

Author

Hugo Pfoertner, Jul 04 2018

Extensions

a(7)-a(10) from Anton Pirogov, Jun 21 2026

Status

approved