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A137939 Number of 5-way intersections in the interior of a regular 6n-gon. 4

%I

%S 0,0,54,24,180,216,546,336,648,720,990,936,1404,2352,1890,1824,2448,

%T 2592,3078,3720,4284,3960,4554,4464,5400,5616,6318,7896,7308,7560,

%U 8370,8256,9504,9792,11550

%N Number of 5-way intersections in the interior of a regular 6n-gon.

%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 <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>

%e a(3) = 54 because there are 54 points in the interior of an 18-gon at which exactly five diagonals meet.

%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. 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.

%K nonn

%O 1,3

%A _Graeme McRae_, Feb 23 2008

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)