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A291857
Number of convex hulls of vertices of a regular n-gon which include the centroid, up to dihedral symmetry.
0
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
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
KEYWORD
nonn
AUTHOR
J. Stauduhar, Sep 04 2017
EXTENSIONS
More terms from Peter J. Taylor, Sep 07 2017
STATUS
approved