Nearly Isosceles Degree Triangles

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

Write-up coming later

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