I'm trying to find an alternate way of writing this series here: https://en.wikipedia.org/wiki/Minimal_polynomial_of_2cos(2pi/n)
I feel like I could do it if I can come up with a nice formula for:
$$\sum_{k=0}^{n} {{\lfloor{\frac{n + k}{2}}\rfloor}\choose{k}}x^k$$
Any ideas?