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Consider $n$ points equally spaced around the unit circle, joined by all possible combinations of lines to make a complete graph. Let $g(n)$ be the number of triangles formed in the resulting diagram.

For example, $g(3) = 1$, $g(4) = 8$, $g(5) = 35$, $g(6) = 110$.

What is the general formula for $g(n)$?

You can see my initial progress on this puzzle here.

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1 Answer 1

up vote 4 down vote accepted

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The function given there is for equally spaced points on a circle, and links there go to explanations and formulas. One link on that OEIS page is to: for general position points on a circle, and gives the formula C(n+3,6)+C(n+1,5)+C(n,5) surely equivalent to a formula at your relevant blog posting. – Mitch Harris Dec 30 '10 at 17:23
Yep - the difficult part is computing the corrections for multiple intersections. But that OEIS link points to some papers which look like they might clear it up (not sure why I didn't think to check OEIS first...) – Chris Taylor Dec 30 '10 at 17:28

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