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)
R. G. Wilson V, Illustration of a(10)
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
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.
KEYWORD
nonn
AUTHOR
Graeme McRae, Dec 26 2004, revised Feb 23 2008, Feb 26 2008
STATUS
approved