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 A137938 #16 Jul 20 2024 08:35:15 %S A137938 0,12,54,264,420,396,1134,1200,1296,3780,2310,2520,3276,3612,4050, %T A137938 5088,5712,5724,7182,11400,9072,9372,10626,11088,12600,13260,14094, %U A137938 15960,17052,23220,19530,20928,21384,23052,26250,25704,27972,28956,30186,39600,34440,34524 %N A137938 Number of 4-way intersections in the interior of a regular 6n-gon. %C A137938 When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet. %C A137938 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 A137938 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 A137938 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 A137938 Seiichi Manyama, <a href="/A137938/b137938.txt">Table of n, a(n) for n = 1..10000</a> %H A137938 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 A137938 <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a> %F A137938 a(n) = A101364(6*n). - _Seiichi Manyama_, Jul 20 2024 %e A137938 a(3)=54 because there are 54 points in the interior of an 18-gon at which exactly four diagonals intersect. %Y A137938 Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon. %Y A137938 Cf. A006561: number of intersections of diagonals in the interior of regular n-gon. %Y A137938 Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon. %Y A137938 Cf. A101364: number of 4-way intersections in the interior of a regular n-gon. %Y A137938 Cf. A101365: number of 5-way intersections in the interior of a regular n-gon. %Y A137938 Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon %K A137938 nonn %O A137938 1,2 %A A137938 _Graeme McRae_, Feb 23 2008 %E A137938 More terms from _Seiichi Manyama_, Jul 20 2024