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.

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

Author

Andrew Howroyd, Mar 01 2024

Keywords

• nonn

Comments

Achiral means that the polygon has an axis of reflective symmetry.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A283184, A370766, A370767, A370768.

Cf. A231091 (stars up to rotation), A370459 (up to rotation and reflection).

Programs

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)))

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.