I have computed the sequence of unique distances a point can be from the origin in a $2$-D random walk, and for the numbers $1$ thru $14$, the number of unique distances are: $$\begin{array}{c} 1 & 3 & 3 & 6 & 6 & 10 & 9 & 15 & 14 & 20 & 19 & 27 & 23 & 32\end{array}.$$
the sequence $1, 3 , 3, 6, 6, \ldots$. represents that after a $1$ step, $2$-dimensional random walk there is only one possible distance from the origin, which is obviously a distance of $1$. For a two step random walk, the possible distances from the origin are $0$, $\sqrt{2}$, and $2$ so there are three unique distances for a two-step walk.
I'm curious to know why that number decrease when one extra step is added. I would have expected it to be always increasing.
There is kind of a pattern here, but nothing obvious. Any insight?
It helps to separate the even terms from the odd terms. The even terms start $$1, 3, 6, 10, 15, 20, 27, 34, 42, 51, 61, 71, 83, 94, 106, 120, 135, 148, 165, 180, 198, 216$$ and are found at OEIS A047800. No formula is given, but there are a few references and a conjecture that they grow about as $0.79 \frac {n^2}{\sqrt{\log(n)}}$. Note that their $n$ corresponds to your $2n$ The odd term series is unknown to OEIS. They are lower than the preceding even term because there are more Pythagorean triangles. The primitive ones always have one odd and one even leg, which will appear with an odd number of steps.