The binomial coefficients T(n)=n(n+1)/2 are called triangular numbers because T(n) points can be arranged into a triangular array with n points on a side. For example, T(4)=10 points can be arranged in the familiar pattern of bowling pins. Call this triangular array with n points on a side Array(n).
If we count the number of triples of points in Array(n) which are vertices of equilateral triangles with sides parallel to those of the whole array, we get the binomial coefficient (n+1)n(n-1)/6: a nice formula in closed form (i.e., no sum of stuff).
What surprised me when I dared look at it is that if we count the number E(n) of ALL triples of points in Array(n) which are vertices of equilateral triangles, we also get a nice formula.
What is that nice formula for E(n)?
(In reply to re: Not a nice formula
Quite correct. There were a few triangles I missed. Perhaps you helped out a little too much. :)
T(n) = C(n+1,n-1)
(n+1)n(n-1)/6 = C(n+2,n-1)
The final series you describe is E(n) = C(n+3,n-1) which simplifies as
E(n) = (n+3)(n+2)(n+1)n/24