cp's OEIS Frontend

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.

%I A231091 #28 Mar 01 2024 14:23:36
%S A231091 0,0,0,0,1,1,5,27,175,1533,14361,151575,1735869,21594863,289365383,
%T A231091 4158887007,63822480809,1041820050629,18027531255745,329658402237171,
%U A231091 6352776451924233,128686951765990343,2733851297673484765,60781108703102022027,1411481990523638719737
%N 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.
%C A231091 For polygons in general see A000939 and A000949, and especially the Golomb-Welch reference. - _N. J. A. Sloane_, Nov 21 2013
%H A231091 Andrew Howroyd, <a href="/A231091/b231091.txt">Table of n, a(n) for n = 1..200</a>
%H A231091 Stewart Gordon, <a href="/A231091/a231091.svg">The five possible stars for n=7</a> (SVG file)
%H A231091 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unicursal_hexagram">Unicursal hexagram</a>
%F A231091 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 A231091 For n=5, only solution is the regular pentagram.
%e A231091 For n=6, only solution is the unicursal hexagram (see Wikipedia link).
%e A231091 For n=7, two regular heptagrams and three irregular forms are possible.
%o A231091 (PARI) \\ Requires a370068 from A370068, b(n) is A283184.
%o A231091 b(n)={subst(serlaplace(polcoef((1 - x)/(1 + (1 - 2*y)*x + 2*y*x^2) + O(x*x^n), n)), y, 1)}
%o A231091 a(n)={(if(n%2==0 && n > 2, b(n/2-1)/2) + a370068(n))/2} \\ _Andrew Howroyd_, Mar 01 2024
%Y A231091 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 A231091 Cf. A283184.
%K A231091 nonn,nice
%O A231091 1,7
%A A231091 _Stewart Gordon_, Nov 03 2013
%E A231091 a(15) onwards from _Andrew Howroyd_, Feb 23 2024