Generate Aztec triangle of size n automatically?

In the paper titled "Perfect Matchings of Cellular Graphs" by Mihai Ciucu, the Aztec triangle of size n (n= 1, 2, 3, 4, 5, ...) is equivalent to a triangular grid of n1 squares (n1 = 1, 4, 9, 16, 25, ...) .

See the following example: image

Is there a automatical way to generate such patterns? (additionally, is it possiable to get the connected points into a list?)

Also are there some general method to generate Aztec diamond of order n (not just triangle)?

Thank you very much!