Integer Sided Equiangular Hexagons

Let $H (n)$ be the number of distinct integer sided equiangular convex hexagons with perimeter not exceeding $n$ .
Hexagons are distinct if and only if they are not congruent.

You are given $H (6) = 1$ , $H (12) = 10$ , $H (100) = 31248$ .
Find $H (55106)$ .

Equiangular hexagons with perimeter not exceeding $12$

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.