In the above question, I choose terminology from communication engineering, but I will precisely define all the involved quantities below. It is a geometric question and I am not sure if it is trivial, but I was not able to figure it out.
Let $\mathcal{X} = \{(a,b) : a,b \in \{\pm 1, \pm 3, ..., \pm 2^i-1 \} \}$ be a set of points in two dimensions, where $i$ is a positive integer. We call this a square QAM (quadrature amplitude modulation) constellation. The number of points in the set is $|\mathcal{X}| = 4^i$. Examples of this set for $i=1$, $2$, and $3$ can be found by doing a google picture search for "quadrature amplitude modulation". Alternatively, we could define the above set as a coset of the two-dimensional integer lattice, intersected with an appropriate squared bounding region.
Given any constellation point $(x,y) \in \mathcal{X}$, we call $x^2 + y^2$ the squared amplitude level. The question is simple: How many distinct amplitude levels exist in $\mathcal{X}$ for an arbitrary $i$?
My confusion arises from the following: Let for example $i=3$. When I combine all the different combinations of odd integers involved in the set construction, I get $4+3+2+1 = 10$ amplitude levels. However, $5^2+5^2 = 1^2 + 7^2$ gives the same level. So it turns out that the answer here is $9$ and not $10$. I don't know how to generalize this to arbitrary $i$.
Thank you in advance!