Let a, b and c be positive numbers.
Let W, X, Y, Z be four collinear points where |WX| = a, |XY| = b, |YZ| = c and |WZ| = a + b + c.
Let Cin be the circle having the diameter XY.
Let Cout be the circle having the diameter WZ.
The triplet (a, b, c) is called a necklace triplet if you can place k 3 distinct circles C1, C2, ..., Ck such that:
For example, (5, 5, 5) and (4, 3, 21) are necklace triplets, while it can be shown that (2, 2, 5) is not.
Let T(n) be the number of necklace triplets (a, b, c) such that a, b and c are positive integers, and b n. For example, T(1) = 9, T(20) = 732 and T(3000) = 438106.
Find T(1 000 000 000).