For any non-negative integer $k$, $x^2+y^2=5^k$ has $4(k+1)$ integer solutions.
For example, here are 3 circles for $k=1,2,3$. The blue dots represent the vertices (integer solutions), then connect the dots to form polygons (dash lines). The red dots represent the points in-between the vertices.
Tabulated the number of In-between Points using this formula $\text{gcd}(|x_2 - x_1|, |y_2 - y_1|) - 1$ for each adjacent vertices $(x_1,y_1),(x_2,y_2)$ and summed them up.
k Vertices 4(k+1) In-between Points Total polygon boundary points
1 8 4 12
2 12 0 12
3 16 28 44
4 20 40 60
5 24 116 140
6 28 144 172
7 32 524 556
8 36 1272 1308
9 40 3028 3068
10 44 5984 6028
11 48 10780 10828
12 52 23440 23492
13 56 63444 63500
14 60 117480 117540
Questions
Is there a pattern or formula to calculate the number of In-between Points for a given $k$?
Notice for $k=2$, there are $0$ In-between Points. Do other values of $k$ exist that have no In-between Points?
I've tried random online sequence solvers to predict the next number of In-between Points and OEIS, but no luck. :(