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A036403 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). 1
1, 126, 3927, 33156, 97115, 641916, 537607, 4222280, 1744695, 20962830, 4003241, 42626916 (list; graph; refs; listen; history; text; internal format)



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).


Mohammad K. Azarian, A Trigonometric Characterization of Equilateral Triangle, Problem 336, Mathematics and Computer Education, Vol. 31, No. 1, Winter 1997, p. 96. Solution published in Vol. 32, No. 1, Winter 1998, pp. 84-85.

Mohammad K. Azarian, Equating Distances and Altitude in an Equilateral Triangle, Problem 316, Mathematics and Computer Education, Vol. 28, No. 3, Fall 1994, p. 337. Solution published in Vol. 29, No. 3, Fall 1995, pp. 324-325.

Computed by Ilan Mayer (ilan(AT)isgtec.com).


Table of n, a(n) for n=1..12.


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.


Cf. A006600.

Sequence in context: A285921 A086024 A285857 * A286976 A186816 A292882

Adjacent sequences:  A036400 A036401 A036402 * A036404 A036405 A036406




Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)


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

a(9)-a(12) from Jon E. Schoenfield, Oct 26 2008



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Last modified September 28 14:44 EDT 2022. Contains 357073 sequences. (Running on oeis4.)