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%I #18 Feb 16 2018 20:23:57
%S 0,1,8,20,60,112,208,216,480,660,864,1196,1568,2250,2464,2992,3924,
%T 4332,5160,8148,7040,8096,10560,10600,12064,15552,15288,17052,25320,
%U 21080,23360,30360,28288,30940,36288,36852,40128,50076,47120,50840,67620
%N In the interior of a regular 2n-gon with all diagonals drawn, the number of points where exactly three diagonals intersect.
%C When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.
%C 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.
%C 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.
%C 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."
%H Graeme McRae, Feb 23 2008, <a href="/A101363/b101363.txt">Table of n, a(n) for n = 2..105</a>
%H M. F. Hasler, <a href="/A006561/a006561.html">Interactive illustration of A006561(n)</a>
%H B. Poonen and M. Rubinstein, <a href="http://arXiv.org/abs/math.MG/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
%H B. Poonen and M. Rubinstein, <a href="http://dx.doi.org/10.1137/S0895480195281246">Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156 (1998). [Copy on SIAM web site]
%H B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">The number of intersection points made by the diagonals of a regular polygon</a>, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998). [Copy on B. Poonen's web site]
%H B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.m">Mathematica programs for A006561 and related sequences</a>
%H M. Rubinstein, <a href="/A006561/a006561_3.pdf">Drawings for n=4,5,6,...</a>
%H N. J. A. Sloane, <a href="/A006561/a006561_4.pdf">Illustrations of a(8) and a(9)</a>
%H R. G. Wilson V, <a href="/A006561/a006561_1.pdf">Illustration of a(10)</a>
%H <a href="/index/Pol#Poonen">Index entry for Sequences formed by drawing all diagonals in regular polygon</a>
%e 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.
%Y Cf. A006561, A007678, A101364, A101365
%Y A column of A292105.
%Y Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.
%Y Cf. A006561: number of intersections of diagonals in the interior of regular n-gon
%Y Cf. A292104: number of 2-way intersections in the interior of a regular n-gon
%Y Cf. A101364: number of 4-way intersections in the interior of a regular n-gon
%Y Cf. A101365: number of 5-way intersections in the interior of a regular n-gon
%Y Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon
%Y Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon.
%K nonn
%O 2,3
%A _Graeme McRae_, Dec 26 2004, revised Feb 23 2008, Feb 26 2008
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