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 $C_{in}$ be the circle having the diameter $XY$.
Let $C_{out}$ be the circle having the diameter $WZ$.

The triplet $(a,b,c)$ is called a necklace triplet if you can place $k\ge 3$ distinct circles $C_1,C_2,\dots ,C_k$ such that:

• $C_i$ has no common interior points with any $C_j$ for $1\le i,j\le k$ and $i\ne j$,
• $C_i$ is tangent to both $C_{in}$ and $C_{out}$ for $1\le i\le k$,
• $C_i$ is tangent to $C_{i+1}$ for $1\le i<k$, and
• $C_k$ is tangent to $C_1$.

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 $b\le n$. For example, $T(1)=9$, $T(20)=732$ and $T(3000)=438106$.

Find $T(1000000000)$.