A231091 Number of distinct (modulo rotation) unicursal star polygons (not necessarily regular, no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon.

0, 0, 0, 0, 1, 1, 5, 27, 175, 1533, 14361, 151575, 1735869, 21594863, 289365383, 4158887007, 63822480809, 1041820050629, 18027531255745, 329658402237171, 6352776451924233, 128686951765990343, 2733851297673484765, 60781108703102022027, 1411481990523638719737

Offset

1,7

Comments

For polygons in general see A000939 and A000949, and especially the Golomb-Welch reference. - N. J. A. Sloane, Nov 21 2013

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Stewart Gordon, The five possible stars for n=7 (SVG file)

Wikipedia, Unicursal hexagram

Formula

a(n) = (A370068(n) + A283184(n/2-1)/2)/2 for even n >= 4; a(n) = A370068(n)/2 for odd n. - Andrew Howroyd, Feb 24 2024

Example

For n=5, only solution is the regular pentagram.
For n=6, only solution is the unicursal hexagram (see Wikipedia link).
For n=7, two regular heptagrams and three irregular forms are possible.

Prog

(PARI) \\ Requires a370068 from A370068, b(n) is A283184.
b(n)={subst(serlaplace(polcoef((1 - x)/(1 + (1 - 2*y)*x + 2*y*x^2) + O(x*x^n), n)), y, 1)}
a(n)={(if(n%2==0 && n > 2, b(n/2-1)/2) + a370068(n))/2} \\ Andrew Howroyd, Mar 01 2024

Crossrefs

Cf. A000939 (if edges may join adjacent vertices), A000940, A002816 (rotations and reflections counted separately), A326411, A370459 (up to rotations and reflections), A370068 (directed edges).

Cf. A283184.

Sequence in context: A091101 A185622 A225309 * A205774 A367048 A326094

Adjacent sequences: A231088 A231089 A231090 * A231092 A231093 A231094

Keyword

nonn,nice

Author

Stewart Gordon, Nov 03 2013

Extensions

a(15) onwards from Andrew Howroyd, Feb 23 2024

Status

approved