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A370769
Number of achiral unicursal star polygons (no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon.
2
0, 0, 0, 0, 1, 1, 5, 11, 49, 123, 521, 1583, 6581, 23239, 95509, 384771, 1570265, 7106995, 28869825, 145034327, 587270877, 3242792607, 13100475021, 78866628011, 318067071169, 2073381189259, 8350998470777, 58602568320255, 235794888434053, 1772311322357623
OFFSET
1,7
COMMENTS
Achiral means that the polygon has an axis of reflective symmetry.
LINKS
FORMULA
a(2*n+1) = A370766(n)/2 - A370768(n-1) for n >= 1.
a(2*n) = (A370766(n-1)/2 - A370768(n-2) + A370766(n)/4 - A370768(n-1) + A283184(n-1)/2)/2 for n >= 2.
PROG
(PARI)
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 - 3*x)/(8*(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, Ro(n\2), Re(n/2)))
CROSSREFS
Cf. A231091 (stars up to rotation), A370459 (up to rotation and reflection).
Sequence in context: A176609 A041213 A142238 * A149514 A149515 A149516
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Mar 01 2024
STATUS
approved