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A101363 In the interior of a regular 2n-gon with all diagonals drawn, the number of points where exactly three diagonals intersect. 6
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

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 A007016

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

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Last modified October 20 23:39 EDT 2018. Contains 316405 sequences. (Running on oeis4.)