Search: id:A101364
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%I A101364
%S A101364 0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,54,0,0,0,0,0,264,0,0,0,0,0,420,0,0,0,0,
%T A101364 0,396,0,0,0,0,0,1134,0,0,0,0,0,1200,0,0,0,0,0,1296,0,0,0,0,0,3780,0,0,
%U A101364 0,0,0,2310,0,0,0,0,0,2520,0,0,0,0,0,3276,0,0,0,0,0,3612,0,0,0,0,0,4050
%N A101364 In the interior of a regular n-gon with all diagonals drawn, the number 
               of points where exactly four diagonals intersect.
%C A101364 When n is odd, there are no intersections in the interior of an n-gon 
               where more than 2 diagonals meet.
%C A101364 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 A101364 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 A101364 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 A101364 Graeme McRae (g_m(AT)mcraefamily.com), Feb 23 2008, <a href="b101364.txt">
               Table of n, a(n) for n = 3..210</a>
%H A101364 <a href="Sindx_Pol.html#Poonen">Sequences formed by drawing all diagonals 
               in regular polygon</a>
%e A101364 a(18)=54 because inside a regular 18-gon there are 54 points where exactly 
               four diagonals intersect.
%Y A101364 Cf. A006561, A007678.
%Y A101364 Cf. A000332: C(n, 4) = number of intersection points of diagonals of 
               convex n-gon.
%Y A101364 Cf. A006561: number of intersections of diagonals in the interior of 
               regular n-gon
%Y A101364 Cf. A101363: number of 3-way intersections in the interior of a regular 
               2n-gon
%Y A101364 Cf. A101365: number of 5-way intersections in the interior of a regular 
               n-gon
%Y A101364 Cf. A137938: number of 4-way intersections in the interior of a regular 
               6n-gon
%Y A101364 Cf. A137939: number of 5-way intersections in the interior of a regular 
               6n-gon
%Y A101364 Sequence in context: A004022 A083344 A063863 this_sequence A104203 A004012 
               A072837
%Y A101364 Adjacent sequences: A101361 A101362 A101363 this_sequence A101365 A101366 
               A101367
%K A101364 nonn
%O A101364 3,10
%A A101364 Graeme McRae (g_m(AT)mcraefamily.com), Dec 26 2004, revised Feb 23 2008

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