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A230281 The least possible number of intersection points of the diagonals in the interior of a convex n-gon with all diagonals drawn. 3
0, 1, 5, 13, 29, 49 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

COMMENTS

Perhaps a(9) = 94.

After removing two points from the regular 12-gon, that is, removing the corresponding points at 12 o'clock and 2 o'clock, there will be only 157 intersection points of the diagonals, it is less than 161, which is the number of intersections of diagonals in the interior of regular 10-gon. So, a(10) <= 157 < 161 = A006561(10). - Guang Zhou, Jul 27 2018

The greatest possible number of intersection points occurs when each set of four vertices gives diagonals with a unique intersection point.  Thus, a(n) <= binomial(n,4) = A000332(n). - Michael B. Porter, Jul 30 2018

LINKS

Table of n, a(n) for n=3..8.

Nathaniel Johnston, Illustration of a(4), a(5), and a(6)

Vladimir Letsko, Mathematical Marathon at VSPU, Problem 102 (in Russian)

Vladimir Letsko, Illustration of a(8) = 49 (the regular octagon provides another example)

V. A. Letsko and M. A. Voronina, Classification of convex polygons, Grani Poznaniya, 1(11), 2011 (in Russian).

V. A. Letsko and M. A. Voronina, Illustration of a(7) = 29

B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).

B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, arXiv version, which has fewer typos than the SIAM version.

EXAMPLE

a(6) = 13 because the number of intersection points of the diagonals in the interior of convex hexagon is equal to 13 if 3 diagonals meet in one point, and this number cannot be less than 13 for any hexagon.

CROSSREFS

Cf. A000332, A006561, A160860.

Sequence in context: A270106 A304904 A095085 * A093836 A000328 A272750

Adjacent sequences:  A230278 A230279 A230280 * A230282 A230283 A230284

KEYWORD

nonn,more,nice

AUTHOR

Vladimir Letsko, Oct 15 2013

STATUS

approved

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Last modified October 18 13:43 EDT 2018. Contains 316321 sequences. (Running on oeis4.)