Viewpoint 1
To generate the even numbers $<n$ the smallest amount of numbers we need about $O (\sqrt{n})$ (by summing only $2$ elements)
Hence, consider the series:
$$ ( \sum_{r=1}^n x^{b_r})^2 = \sum_{i=1}^n \sum_{j=1}^n x^{b_i + b_j}= \sum_{r=2}^n c_{r} x^{2r} $$
where $b_i$ is an arbitrary natural number.
For example for $n= 57$
$$ (x^2 + x^4 + x^8 + x^{12} + x^{16} + x^{18} + x^{36}+ x^{38})^2= \sum_{r=1}^{28} c_r x^{2r} $$ and then taking the square root and the limit:
$$ \lim_{x \to 1^-} ( \sum_{r=1}^n x^{b_r}) = \sqrt{\sum_{r=2}^\infty c_{r} x^{2r}} = \text{number of terms}$$
Proceeding in our example:
$$ \lim_{x \to {1^-}}(x^2 + x^4 + x^8 + x^{12} + x^{16} + x^{18} + x^{36}+ x^{38})^2= 8 \approx \sqrt{57} $$
Hence, to generate all the even numbers:
$$ \lim_{n \to \infty }\lim_{x \to 1^-} ( \sum_{r=1}^n x^{b_r})^2 \sim O(\sqrt{n})$$
Viewpoint 2
However I do not think we can say that is the case to generate all the even numbers.
We need: Let the series which generates all even numbers be
$$ (\lim_{n \to \infty} \sum_{r=1}^n x^{a_r})^2= \sum_{i=1}^n \sum_{j=1}^n x^{a_i + a_j} = \sum_{r=2}^\infty c_{r} x^{2r} $$
where $a_i$ is an arbitrary natural number. Assuming Goldbach's conjecture is true we can use the following we can use the following example:
$$ (\lim_{n \to \infty} \sum_{r=1}^n x^{p_r})^2 = \sum_{r=2}^\infty c_{r} x^{2r} $$
Where $p_r$ is the $r$'th prime
$$ \lim_{n \to \infty} (\sum_{r=1}^n x^{p_r})^2 = \sum_{r=2}^\infty c_{r} x^{2r} $$
And the number of terms would approach required to generate the even powered series:
$$ \implies \lim_{n \to \infty} (\sum_{r=1}^n x^{a_r}) = \sqrt{\sum_{r=2}^\infty c_r x^{2r} }= \text{number of terms}\sim \alpha(x) $$
$x \to 1^-$
Again proceeding with our example:
$$ \lim_{n \to \infty} (\sum_{r=1}^n x^{p_r})\sim \frac{1}{(x-1)\ln(1-x)} $$
where $x \to 1^-$
Where $ \alpha(x) $ would be the amount to generate all the even numbers. Hence, the number of elements we need are:
$$ \implies \lim_{n \to \infty} \sum_{r=1}^n a_r x^r \sim \alpha(x) $$
where $x \to 1^-$
Question
Which viewpoint is correct? and why?
P.S: This is related to Reformulation of Goldbach's Conjecture as optimization problem correct? however I did not have enough space in the comments to dispute my misconception/concept.
Point of Dispute (Edit)
How does one extend the finite case $< n $ to $n \to \infty $
Viewpoint 2 seems to suggest they should have no connection.
Given $n$, choose $K$ even such that $K^2>2n$, and use the numbers $$2,4,6,8,\dots,K-4,K-2,K,2K,3K,4K,\dots,(K/2)K$$ That uses about $K$ numbers, where $K$ is about $\sqrt2\sqrt n$.
One can do better than that, but that's enough to show you that $c\sqrt n$ can be achieved for some constant $c$.
Much, much more information is available by following my comment and searching for $$\rm additive\ basis\ of\ order\ 2$$