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%I #14 Jul 19 2024 13:27:23
%S 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,54,0,0,0,0,0,24,0,0,0,0,0,180,0,0,0,0,
%T 0,216,0,0,0,0,0,546,0,0,0,0,0,336,0,0,0,0,0,648,0,0,0,0,0,720,0,0,0,
%U 0,0,990,0,0,0,0,0,936,0,0,0,0,0,1404,0,0,0,0,0,2352,0,0,0,0,0,1890,0,0,0,0
%N In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly five 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 except the center.
%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 except the center.
%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 interior point other than the center 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 (all points interior excluding the center)."
%H Seiichi Manyama, <a href="/A101365/b101365.txt">Table of n, a(n) for n = 3..10000</a> (terms 3..210 from Graeme McRae)
%H <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>
%e a(18)=54 because inside a regular 18-gon there are 54 points (3 on each radius) where exactly five diagonals intersect.
%Y A column of A292105.
%Y Cf. A006561, A007678.
%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. A101363: number of 3-way intersections in the interior of a regular 2n-gon.
%Y Cf. A101364: number of 4-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 3,16
%A _Graeme McRae_, Dec 26 2004, revised Feb 23 2008