Problem 428
Necklace of circles
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:
- Ci has no common interior points with any Cj for 1 ≤ i, j ≤ k and i ≠ j,
- Ci is tangent to both Cin and Cout for 1 ≤ i ≤ k,
- Ci is tangent to Ci+1 for 1 ≤ i < k, and
- Ck is tangent to C1.
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).
圆圈项链
取a，b，c均为正整数。
取共线的四点W，X，Y，Z，满足|WX| = a，|XY| = b，|YZ| = c而|WZ| = a + b + c。
圆Cin是以XY为直径的圆。
圆Cout是以WZ为直径的圆。
三元组(a, b, c)被称为项链三元组，如果存在k ≥ 3个不同的圆C1，C2，……，Ck满足：
- 对于任意1 ≤ i, j ≤ k且i ≠ j，Ci和Cj没有公共内部点。
- 对于任意1 ≤ i ≤ k，Ci和Cin与Cout均相切。
- 对于任意1 ≤ i < k，Ci和Ci+1相切。
- Ck和C1相切。
例如，(5, 5, 5)和(4, 3, 21)都是项链三元组，而(2, 2, 5)则不是。
记T(n)是项链三元组(a, b, c)的数量，其中a，b，c均为正整数，且b ≤ n。例如，T(1) = 9，T(20) = 732以及T(3000) = 438106。
求T(1 000 000 000)。