A370459 Number of unicursal stars with n vertices.

0, 0, 1, 1, 5, 19, 112, 828, 7441, 76579, 871225, 10809051, 144730446, 2079635889, 31912025537, 520913578812, 9013780062785, 164829273635749, 3176388519597555, 64343477504391475, 1366925655386979893, 30390554390984325019, 705740995420852895453

Offset: 3

Author

Adam M. Scherlis, Feb 19 2024

Keywords

• nonn

Comments

A unicursal star is a closed loop formed by diagonals of a regular n-gon.

These are Hamiltonian cycles on the graph complement of the n-cycle.

Allowing polygon diagonals, but not sides, is equivalent to requiring every edge to cross at least one other edge.

These are counted up to rotation and reflection, i.e., modulo dihedral symmetry of the n-gon.

Inspired by a unicursal dodecagram drawn by Gordon FitzGerald (see links).

Examples

			For n=5, there is only the regular pentagram {5/2}.
For n=6, there is only the unicursal hexagram.
For n=7, in addition to the two regular heptagrams {7/2} and {7/3}, there are three nontrivial unicursal heptagrams represented by:
 (0, 2, 4, 1, 6, 3, 5, 0)
 (0, 2, 5, 1, 3, 6, 4, 0)
 (0, 2, 5, 1, 4, 6, 3, 0).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..200
• Gordon FitzGerald, illustration of a unicursal dodecagram.
• Wikipedia, Unicursal hexagram.

Crossrefs

Cf. A000940 (polygon sides allowed).

Cf. A055684 (cases with dihedral symmetry only).

Cf. A002816 (rotations and reflections counted separately).

Cf. A231091 (up to rotations only), A370769 (achiral).

Programs

PARI

\\ Requires a370068 from A370068.
Ro(n)=-(-1)^n + subst(serlaplace(polcoef(((1 - x)^2)/(2*(1 + x)*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
Re(n)=subst(serlaplace(polcoef((1 - x - 2*x^2)/(4*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
a(n)={if(n<3, 0, (if(n%2, 2*Ro(n\2), Re(n/2)) + a370068(n))/4)} \\ Andrew Howroyd, Mar 01 2024

Formula

a(n) = (A231091(n) + A370769(n))/2. - Andrew Howroyd, Mar 06 2024

Extensions

a(14) onwards from Andrew Howroyd, Feb 26 2024