>So, you're looking for when a series becomes two triangles where every unit of the triangle is a square.
I think this thing from Grid Patterns could potentially be a relevant crumb. Notice the 4, 4+4, 4+4+4 thing. I'm not sure where you'd expand this, but it's the only instance I can remember of a triangle with every unit being a square.
If you take a square and look for it as a b value in (0,n), the difference in n values from record to record is equal to (2x)+(4T(x)).
e.g. 9*9=81
These are the records we will get (since there can't be opposite parity between a and b as it would create an invalid n, we will only see the square of all odd numbers below 9 in a[t]):
{0:2:63:14:49:81}
{0:8:45:20:25:81}
{0:18:27:18:9:81}
{0:32:9:8:1:81}
The difference in n values can be expressed as follows:
8 - 2 = 6, 6=2+4
18 - 8 = 10, 10=2+4+4
32 - 18 = 14, 14=2+4+4+4