A101363 - OEIS

A101363

In the interior of a regular 2n-gon with all diagonals drawn, the number of points where exactly three diagonals intersect.

0, 1, 8, 20, 60, 112, 208, 216, 480, 660, 864, 1196, 1568, 2250, 2464, 2992, 3924, 4332, 5160, 8148, 7040, 8096, 10560, 10600, 12064, 15552, 15288, 17052, 25320, 21080, 23360, 30360, 28288, 30940, 36288, 36852, 40128, 50076, 47120, 50840, 67620

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

OFFSET

2,3

COMMENTS

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.

When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet.

When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet.

I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no points where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points."

LINKS

Seiichi Manyama, Table of n, a(n) for n = 2..10000 (terms 2..105 from Graeme McRae)

M. F. Hasler, Interactive illustration of A006561(n)

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, which has fewer typos than the SIAM version.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156 (1998). [Copy on SIAM web site]

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). [Copy on B. Poonen's web site]

B. Poonen and M. Rubinstein, Mathematica programs for A006561 and related sequences

M. Rubinstein, Drawings for n=4,5,6,...

N. J. A. Sloane, Illustrations of a(8) and a(9)

N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]

R. G. Wilson V, Illustration of a(10)

Index entry for Sequences formed by drawing all diagonals in regular polygon

EXAMPLE

a(6)=60 because inside a regular 12-gon there are 60 points (4 on each radius and 1 midway between radii) where exactly three diagonals intersect.

CROSSREFS

Cf. A006561, A007678, A101364, A101365

A column of A292105.

Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.