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

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

Graeme McRae, Feb 23 2008, Table of n, a(n) for n = 2..105

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)

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.

Cf. A006561: number of intersections of diagonals in the interior of regular n-gon

Cf. A292104: number of 2-way intersections in the interior of a regular n-gon

Cf. A101364: number of 4-way intersections in the interior of a regular n-gon

Cf. A101365: number of 5-way intersections in the interior of a regular n-gon

Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon

Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon.

Sequence in context: A212758 A179756 A238507 * A003685 A066011 A333156

Adjacent sequences: A101360 A101361 A101362 * A101364 A101365 A101366

Keyword

nonn

Author

Graeme McRae, Dec 26 2004, revised Feb 23 2008, Feb 26 2008

Status

approved