Out of n straight lines whose lengths are 1, 2, 3,...,n inches respectively. Prove that the number of ways in which four may be chosen which will form a quadrilateral in which a circle may be inscribed is .
A quadrilateral with side lengths can be inscribed by a circle if and only if . So to restate the problem in much simpler form, given the set of numbers , in how many ways can four distinct numbers be chosen from this set satisfying without regard to order?
Denote the sequence in question by which we are conjecturing is