# Combinatorics and upper bound

Consider the set $$\{\lambda = \alpha _1^2 + \alpha _2^2 + \alpha _3 ^2 : \alpha _j \in \mathbb{Z}, \, j=1,2,3 \}$$ of real numbers. Assume I order this set so that $\lambda _1 \le \lambda _2 \le \lambda _3 \le \cdots$ in increasing order. Of course every $\lambda _j$ will be repeated a few times (except for 0). I want to obtain a bound of the form $$\lambda _j \le \varphi (j)$$
for some function $\varphi$. It probably doesn't have to be a sharp bound but the sharper the better, of course. At least I'm looking for a polynomial bound of the form $\lambda _j \le Cj^k$ for some positive integer $k$.

The number of sequence elements $\lambda_j$ that are less than some $x$ is at least $(2\sqrt{x/3}-1)^3$, by considering $\alpha_1,\alpha_2,\alpha_3\in[-\lfloor\sqrt{x/3}\rfloor,\lfloor\sqrt{x/3}\rfloor]$. Therefore $\lambda_{\lceil(2\sqrt{x/3}-1)^3\rceil} \le x$, which implies that $\lambda_j \le Cj^{2/3}$ for some constant $C>0$.