What are practical ways to simplify/represent $0,n,0,n,0,n...$ sequences to make a power series more efficient?

Question

Suppose I have a series in the form of $$ \sum_{k=0}^{n}((-1)^n+1)x^n$$

This will look like $S_n = 2+0+2x^2+0+2x^4+0...$

How do I re-index the sum so that either the $((-1)^n+1)$ term is no longer two terms, but just one term as a function of n so that I can pair it with $z^n$ such that; if $u = ((-1)^{n}+1)$, then $S_n = (uz)^{n}$? Do I need some kind of step function or is there a less awkward way to write it, using only functions that need not be constrained to descrete number systems (f(n)=2n is instantly interpolable to the complex number system)

And, would it be correct to just re-rewrite it as $2 \sum_{k=0}^{n}x^{2n}$?

How does one prove that second form with index notation? Is there a way to do it without induction? I would think whatever method of proof for it should be applicable to other kinds of $a_k$ terms, since $x^{2k}$ is easy to work with.

Answer 1

$$\sum_{k=0}^n ((-1)^k + 1) x^k = 2\sum_{k=0}^{\lfloor n/2 \rfloor} x^{2k}$$

Answer 2

If your notion of efficiency allows for the Iverson bracket ($[p]=1$ if $p$ is true, $0$ if $p$ is false), then \begin{equation} \sum_{k=0}^n ((-1)^k + 1)x^k = 2 \sum_{k=0}^n [2|k] x^k = 2 \sum_{k=0}^n ([2|k] x)^k \end{equation} where $2|k$ is the statement that 2 divides evenly into $k$.