Decomposable Triangles

An integer equilateral triangle is decomposable if it can be split in 2 integer triangles.

For instance, the triangle (8,8,8) can be decomposed as (8,3,7) + (8,5,7)


How many triangles (T) with side $\le {10}^7$ are decomposable?
Which triangle (L) has the largest number of decompositions?(if multiple possibilities, give the largest one)
How many decompositions (D) does it possess?

For instance, the triangle (2592,2592,2592) has 3 decompositions:
- (2592 2457 297)+(2592 2457 2295)
- (2592 2322 702)+(2592 2322 1890)
- (2592 2268 972)+(2592 2268 1620)

Answer format: T,L,D

You are given: 364,960,8 // For triangles with side $\le 1000$

My timing: 20 sec