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%I #13 Jul 07 2023 14:45:29
%S 1,126,3927,33156,97115,641916,537607,4222280,1744695,20962830,
%T 4003241,42626916
%N Number of equilateral triangles whose vertices (whether connected by lines or not) lie at intersection points resulting from drawing lines connecting every pair of vertices of a regular 3n-gon (and extending beyond the polygon).
%C Given a regular 3n-gon, draw a line, extending beyond the polygon, through every pair of vertices; a(n) is the number of distinct equilateral triangles whose vertices lie at three of the resulting intersection points (whether the three points are connected by lines or not).
%D Computed by Ilan Mayer (ilan(AT)isgtec.com).
%e Drawing lines connecting every pair of vertices on a regular hexagon (6-gon) and extending those lines beyond the polygon results in 37 distinct intersection points. Of the 37 * 36 * 35 / 3! = 7770 sets of 3 of those intersection points that could be selected, there are 126 sets of 3 intersection points such that, if the 3 points were connected by line segments, the resulting triangle would be equilateral, so a(2)=126.
%Y Cf. A006600.
%K nonn,nice,more
%O 1,2
%A Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)
%E Added a(5) through a(8), corrected definition and comment and provided example, after receiving clarification Oct 22 2008 from Ilan Mayer (who had originally computed the sequence) regarding its definition. - _Jon E. Schoenfield_, Oct 23 2008
%E a(9)-a(12) from _Jon E. Schoenfield_, Oct 26 2008
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