Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Jul 20 2024 06:48:33
%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 8370,8256,9504,9792,11550,10584,11988,12312,13338,14640,14760,17640,16254,16104,17820,18216,19458,19296,22344,21600
%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 Seiichi Manyama, <a href="/A137939/b137939.txt">Table of n, a(n) for n = 1..10000</a>
%H B. Poonen and M. Rubinstein, <a href="https://arxiv.org/abs/math/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
%H <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>
%F a(n) = A101365(6*n). - _Seiichi Manyama_, Jul 20 2024
%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
%E More terms from _Seiichi Manyama_, Jul 20 2024