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Odd Triplets

Problem 242

Published on Saturday, 25th April 2009, 06:00 am; Solved by 662

Given the set {1,2,...,n}, we define f(n,k) as the number of its k-element subsets with an odd sum of elements. For example, f(5,3) = 4, since the set {1,2,3,4,5} has four 3-element subsets having an odd sum of elements, i.e.: {1,2,4}, {1,3,5}, {2,3,4} and {2,4,5}.

When all three values n, k and f(n,k) are odd, we say that they make
an odd-triplet [n,k,f(n,k)].

There are exactly five odd-triplets with n≤ 10, namely:
[1,1,f(1,1) = 1], [5,1,f(5,1) = 3], [5,5,f(5,5) = 1], [9,1,f(9,1) = 5] and [9,9,f(9,9) = 1].

How many odd-triplets are there with n≤ 1012 ?