A291857 Number of convex hulls of vertices of a regular n-gon which include the centroid, up to dihedral symmetry.

0, 1, 1, 3, 3, 8, 10, 22, 32, 64, 102, 200, 336, 643, 1144, 2170, 3960, 7533, 14022, 26724, 50404, 96349, 183322, 351610, 673044, 1294714, 2489502, 4801854, 9264396, 17912476, 34652962, 67142410, 130182972, 252712216, 490918440, 954571510, 1857413460, 3617082841

Offset

1,4

Comments

When n is even, the convex hull of two vertices which are opposite each other is a diameter of the circumcircle of the n-gon, and is counted as including the centroid. - Peter J. Taylor, Sep 07 2017

Links

Table of n, a(n) for n=1..38.

Formula

a(n) = A000029(n) - 2 - 1/2 Sum_{k=0..floor((n-3)/2)} (2^k + 2^ceiling(k/2)). - Peter J. Taylor, Sep 07 2017

Example

For n = 5 a convex hull needs at least three vertices to contain the centroid of the pentagon. There are two convex hulls of three vertices up to symmetry, of which the "fat" triangle doesn't contain the centroid, and the "thin" triangle does. There is one convex hull of four vertices and one of five vertices up to symmetry, both of which contain the centroid. Therefore a(5) = 3.

Crossrefs

Cf. A000029.

Sequence in context: A123315 A237113 A293937 * A300672 A368726 A052407

Adjacent sequences: A291854 A291855 A291856 * A291858 A291859 A291860

Keyword

nonn

Author

J. Stauduhar, Sep 04 2017

Extensions

More terms from Peter J. Taylor, Sep 07 2017

Status

approved