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A370459
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Number of unicursal stars with n vertices.
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4
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0, 0, 1, 1, 5, 19, 112, 828, 7441, 76579, 871225, 10809051, 144730446, 2079635889, 31912025537, 520913578812, 9013780062785, 164829273635749, 3176388519597555, 64343477504391475, 1366925655386979893, 30390554390984325019, 705740995420852895453
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OFFSET
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3,5
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COMMENTS
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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).
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LINKS
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FORMULA
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EXAMPLE
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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).
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PROG
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(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
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CROSSREFS
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Cf. A000940 (polygon sides allowed).
Cf. A055684 (cases with dihedral symmetry only).
Cf. A002816 (rotations and reflections counted separately).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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