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A370459
Number of unicursal stars with n vertices.
4
0, 0, 1, 1, 5, 19, 112, 828, 7441, 76579, 871225, 10809051, 144730446, 2079635889, 31912025537, 520913578812, 9013780062785, 164829273635749, 3176388519597555, 64343477504391475, 1366925655386979893, 30390554390984325019, 705740995420852895453
OFFSET
3,5
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).
LINKS
FORMULA
a(n) = (A231091(n) + A370769(n))/2. - Andrew Howroyd, Mar 06 2024
EXAMPLE
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).
PROG
(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
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).
Sequence in context: A158615 A321652 A088180 * A206709 A199480 A339079
KEYWORD
nonn
AUTHOR
Adam M. Scherlis, Feb 19 2024
EXTENSIONS
a(14) onwards from Andrew Howroyd, Feb 26 2024
STATUS
approved