Eccoci.

Let $a, b$ and $c$ be the sides of an integer sided triangle with one angle of $120$ degrees, $a \le b \le c$ and $b-a \le 100$.
Let $T(n)$ be the number of such triangles with $c \le n$.
$T(1000)=235$ and $T(10^8)=1245$.
Find $T(10^{100})$.