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%I #57 Aug 07 2018 04:15:31
%S 0,1,5,13,29,49
%N The least possible number of intersection points of the diagonals in the interior of a convex n-gon with all diagonals drawn.
%C Perhaps a(9) = 94.
%C 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
%C 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
%H Nathaniel Johnston, <a href="/A230281/a230281.png">Illustration of a(4), a(5), and a(6)</a>
%H Vladimir Letsko, <a href="http://www-old.fizmat.vspu.ru/doku.php?id=marathon:problem_102">Mathematical Marathon at VSPU, Problem 102</a> (in Russian)
%H Vladimir Letsko, <a href="/A230281/a230281.jpg">Illustration of a(8) = 49</a> (the regular octagon provides another example)
%H V. A. Letsko and M. A. Voronina, <a href="http://grani.vspu.ru/files/publics/1301378772.pdf">Classification of convex polygons</a>, Grani Poznaniya, 1(11), 2011 (in Russian).
%H V. A. Letsko and M. A. Voronina, <a href="/A230281/a230281.pdf">Illustration of a(7) = 29</a>
%H B. Poonen and M. Rubinstein, <a href="https://math.mit.edu/~poonen/papers/ngon.pdf">The number of intersection points made by the diagonals of a regular polygon</a>, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
%H B. Poonen and M. Rubinstein, <a href="https://arxiv.org/abs/math/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG], 1995-2006, arXiv version, which has fewer typos than the SIAM version.
%e 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.
%Y Cf. A000332, A006561, A160860.
%K nonn,more,nice
%O 3,3
%A _Vladimir Letsko_, Oct 15 2013
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