This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137939 #14 Jul 20 2024 06:48:33 %S A137939 0,0,54,24,180,216,546,336,648,720,990,936,1404,2352,1890,1824,2448, %T A137939 2592,3078,3720,4284,3960,4554,4464,5400,5616,6318,7896,7308,7560, %U A137939 8370,8256,9504,9792,11550,10584,11988,12312,13338,14640,14760,17640,16254,16104,17820,18216,19458,19296,22344,21600 %N A137939 Number of 5-way intersections in the interior of a regular 6n-gon. %C A137939 When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet. %C A137939 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 A137939 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 A137939 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 A137939 Seiichi Manyama, <a href="/A137939/b137939.txt">Table of n, a(n) for n = 1..10000</a> %H A137939 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 A137939 <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a> %F A137939 a(n) = A101365(6*n). - _Seiichi Manyama_, Jul 20 2024 %e A137939 a(3) = 54 because there are 54 points in the interior of an 18-gon at which exactly five diagonals meet. %Y A137939 Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.. %Y A137939 Cf. A006561: number of intersections of diagonals in the interior of regular n-gon. %Y A137939 Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon. %Y A137939 Cf. A101364: number of 4-way intersections in the interior of a regular n-gon. %Y A137939 Cf. A101365: number of 5-way intersections in the interior of a regular n-gon. %Y A137939 Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon. %K A137939 nonn %O A137939 1,3 %A A137939 _Graeme McRae_, Feb 23 2008 %E A137939 More terms from _Seiichi Manyama_, Jul 20 2024