Limit points of Nathanson's Lambda sequences

We consider the set $A_{n}=\displaystyle\cup_{j=0}^{\infty}{\varepsilon_{j}(n)\cdot n^j\colon\varepsilon_{j}(n)\in{0,\pm1,\pm2,…,\pm\lfloor{{n}/{2}}\rfloor}} $\mathrm{.}Let$$\mathcal{S}${$\mathcal{A}$}= \bigcup{a \in$\mathcal{A}$ } A_{a}$where$$\mathcal{A}$\subseteq $\mathbb{N}$$\mathrm{.}Wedenoteby$\lambda_{$\mathcal{A}$}(h)$ the smallest positive integer that can be represented as a sum of $h$, and no less than $h$, elements in ${\mathcal{S}}_{\mathcal{A}}$. Nathanson studied the properties of the ${\lambda }_{\mathcal{A}}(h)$-sequence and posed the problem of finding the values of ${\lambda }_{\mathcal{A}}(h)$. When $\mathcal{A}=\{2,i\}$, we represent ${\lambda }_{\mathcal{A}}(h)$ by ${\lambda }_{2,i}(h)$. Only the values ${\lambda }_{2,3}(1)=1$, ${\lambda }_{2,3}(3)=5$, ${\lambda }_{2,3}(3)=21$ and ${\lambda }_{2,3}(4)=150$ are known. In this paper, we extend this result. For any odd $i>1$ and $h\in \{1,2,3\}$, we find the values of ${\lambda }_{2,i}(h)$. Furthermore, for fixed $h\in \{1,2,3\}$, we find the values of ${\lambda }_{2,i}(h)$ that occur infinitely many times as $i$ runs over the odd integers bigger than 1. We call these numbers the $\textit{limit points of Nathanson’s lambda sequences}$.