Playing with another way of structuring the triangles. Not really sure I'm heading in any productive way, but I was curious and wanted to take a look at it. I was a bit inspired by >>9200, but I'm not thinking this is the direction he was thinking of.
This is based on c=3367 with the different cells (3367 = 7x13x37).
This makes me think it's all recursion. That our placement in our triangles for (x+n)^2 is just like what I'm seeing here.
It's all about tiling and what each value contributes. From the images above, we could generate 8 triangles and add a unit, giving us a, somehow, related (x+n)^2. From what I can tell, it could easily be that our f-2 represents a part of another value for another element.