

A101363


In the interior of a regular 2ngon 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
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OFFSET

2,3


COMMENTS

When n is odd, there are no intersections in the interior of an ngon where more than 2 diagonals meet.
When n is not a multiple of 6, there are no intersections in the interior of an ngon where more than 3 diagonals meet.
When n is not a multiple of 30, there are no intersections in the interior of an ngon where more than 5 diagonals meet.
I checked the following conjecture up to n=210: "An ngon 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], 19952006, 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. 135156 (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, 135156 (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 12gon 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 ngon.
Cf. A006561: number of intersections of diagonals in the interior of regular ngon
Cf. A292104: number of 2way intersections in the interior of a regular ngon
Cf. A101364: number of 4way intersections in the interior of a regular ngon
Cf. A101365: number of 5way intersections in the interior of a regular ngon
Cf. A137938: number of 4way intersections in the interior of a regular 6ngon
Cf. A137939: number of 5way intersections in the interior of a regular 6ngon.
Sequence in context: A212758 A179756 A238507 * A003685 A066011 A007016
Adjacent sequences: A101360 A101361 A101362 * A101364 A101365 A101366


KEYWORD

nonn,changed


AUTHOR

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


STATUS

approved



