A231091 - OEIS

%I #28 Mar 01 2024 14:23:36

%S 0,0,0,0,1,1,5,27,175,1533,14361,151575,1735869,21594863,289365383,

%T 4158887007,63822480809,1041820050629,18027531255745,329658402237171,

%U 6352776451924233,128686951765990343,2733851297673484765,60781108703102022027,1411481990523638719737

%N 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.

%C For polygons in general see A000939 and A000949, and especially the Golomb-Welch reference. - _N. J. A. Sloane_, Nov 21 2013

%H Andrew Howroyd, <a href="/A231091/b231091.txt">Table of n, a(n) for n = 1..200</a>

%H Stewart Gordon, <a href="/A231091/a231091.svg">The five possible stars for n=7</a> (SVG file)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Unicursal_hexagram">Unicursal hexagram</a>

%F 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

%e For n=5, only solution is the regular pentagram.

%e For n=6, only solution is the unicursal hexagram (see Wikipedia link).

%e For n=7, two regular heptagrams and three irregular forms are possible.

%o (PARI) \\ Requires a370068 from A370068, b(n) is A283184.

%o b(n)={subst(serlaplace(polcoef((1 - x)/(1 + (1 - 2*y)*x + 2*y*x^2) + O(x*x^n), n)), y, 1)}

%o a(n)={(if(n%2==0 && n > 2, b(n/2-1)/2) + a370068(n))/2} \\ _Andrew Howroyd_, Mar 01 2024

%Y 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).

%Y Cf. A283184.

%K nonn,nice

%O 1,7

%A _Stewart Gordon_, Nov 03 2013

%E a(15) onwards from _Andrew Howroyd_, Feb 23 2024