A230150 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

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.

LINKS

Table of n, a(n) for n=3..23.

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

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.

EXAMPLE

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)

CROSSREFS

Cf. A006522, A160860, A007678.

Sequence in context: A238489 A002537 A295001 * A301109 A301015 A008184