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, 126, 3927, 33156, 97115, 641916, 537607, 4222280, 1744695, 20962830, 4003241, 42626916

Offset: 1

Author

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

Keywords

• nonn
• nice
• more

Comments

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).

Examples

			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.
		

References

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

Crossrefs

Cf. A006600.

Extensions

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