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

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

Offset: 3

Author

Vladimir Letsko, Oct 11 2013

Keywords

• tabf
• nonn

Comments

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.

Examples

			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)
		

Links

• 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

Crossrefs

Cf. A006522, A160860, A007678.

Formula

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.