A230281 The least possible number of intersection points of the diagonals in the interior of a convex n-gon with all diagonals drawn.

0, 1, 5, 13, 29, 49

Offset: 3

Author

Vladimir Letsko, Oct 15 2013

Keywords

• nonn
• more
• nice

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A000332, A006561, A160860.