## Odd Triplets

### Problem 242

Published on Saturday, 25th April 2009, 06:00 am; Solved by 682; Difficulty rating: 75%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` ≤ 10^{12} ?