My idea about this problem comes from my sister's high school math problem, how many quadrilaterals are there in the graph.
This problem requires patience. You need to find squares, general parallelograms and trapezoids respectively. Finally, it comes to $3108$, or you can find out that the general term is $ n(n+1)(7n^2 - n-3)/6$.
(Let the side length of the smallest square be $1$, and $n$ be the side length of the largest square).
Further exploration, when we find the number of parallelograms or triangles in the graph, we can finally get a quartic polynomial about $n$.
In other problems, such as how many parallelograms or triangles are there in the graph, we can still obtain a quartic polynomial about $n$.
These are other pictures, in which the number of triangles, parallelograms, squares, rectangles and quadrilaterals also seems to be about the quartic polynomial of $n$.
So for the number of rectangles or squares or triangles or parallelograms or quadrilaterals in the regular pattern composed of triangles on these planes, I guess the answer is generally a quartic polynomial with a constant term of $0$ about $n$. We only need to find the first few terms, and then Lagrange interpolation can find the general term. Am I right?
My mother tongue is not English, so I use translation software, so there may be some deviation. As a newcomer to MSE, if my post violates the rules, I will delete it
Yes, if you are counting rectangles in a grid then there are four degrees of freedom - length, width, and x, y coordinates. There are four pieces of data that you need to fully specify a rectangle in the grid. When you are counting all of them you are doing a quadruple sum, leading to a quartic. Same with parallelograms.
If you are just counting squares, or equilateral triangles, then you have only three degrees of freedom - size, x, y - so you get a cubic.
The zero constant term comes from the fact that the n=0 grid consisting of just one point contains no shapes.