A230281 -id:A230281 - cp's OEIS frontend

A160860 The least possible number of pieces resulting from cutting a convex n-gon along all its diagonals.

Original entry on oeis.org

1, 4, 11, 24, 47, 80

Offset: 3

Vladimir Letsko, May 29 2009, May 30 2009, Apr 20 2010

It seems that a(9)=137 and a(n) = A007678(n) for all even n.

Vladimir Letsko, Illustration of all cases for number of sides from 3 to 8
Vladimir Letsko, Illustration of all cases for number of sides from 3 to 8 [Cached copy, pdf version only]
Vladimir Letsko, Proof for n = 7 and n = 8 and example for n = 9 (in Russian)
Vladimir Letsko, Proof for n = 7 and n = 8 and example for n = 9 (in Russian). [Cached copy, pdf version only]
V. A. Letsko, M. A. Voronina, Classification of convex polygons, Grani Poznaniya, 1(11), 2011. (in Russian)
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon.

Cf. A006522, A007678, A230281.

A230503 Irregular triangle read by rows: possible number of interior intersection points of the diagonals of an n-sided convex polygon.

Original entry on oeis.org

0, 1, 5, 13, 15, 29, 31, 33, 35, 49, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 70

Offset: 3

Vladimir Letsko, Oct 21 2013

Beginning from number of sides equal to 12 the terms no longer increase between rows. For example, the number of inner diagonal intersection points for the regular 12-gon is fewer than the number of inner diagonal intersection points for regular 11-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| 0
4| 1
5| 5
6| 13, 15
7| 29, 31, 33, 35
8| 49, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 70

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

Cf. A006561, A230281, A000332, A230150.

cp's OEIS Frontend

A160860 The least possible number of pieces resulting from cutting a convex n-gon along all its diagonals.

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