A160860 -id:A160860 - cp's OEIS frontend

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

Vladimir Letsko, Oct 15 2013

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

			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.

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.

Cf. A000332, A006561, A160860.

A230150 Irregular triangle read by rows: Possible numbers of pieces resulting from cutting a convex n-sided polygon along all its diagonals.

Original entry on oeis.org

1, 4, 11, 24, 25, 47, 48, 49, 50, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 137

Offset: 3

Vladimir Letsko, Oct 11 2013

Beginning from number of sides equal to 18 the terms no longer increase between rows. For example, the number of pieces for the regular 18-gon is fewer than the number of pieces for regular 17-gon.

Obviously there exists a number k_0 such that k_0 is not in the sequence and k is in the sequence for all k > k_0.

			The beginning of the irregular triangle is:
3| 1
4| 4
5| 11
6| 24, 25
7| 47, 48, 49, 50,
8| 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91
9| 137 (incomplete)

V.A. Letsko, M.A. Voronina Classification of convex polygons, Grani Poznaniya, 1(11), 2011. (in Russian)
Vladimir Letsko, Mathematical Marathon at vspu, Problem 102 (in Russian)
Vladimir Letsko Illustration of all cases for number of sides from 3 to 8

Cf. A006522, A160860, A007678.

a(n,s_1,...,s_m) = A006522(n) - sum_{k=1}^m s_k*k*(k+1)/2, where m = floor(n/2)-2 and s_k denotes number of inner points in which exactly k+2 diagonals are intersected.

cp's OEIS Frontend

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

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