Tight universal triangular forms

For a subset $S$ of nonnegative integers and a vector $\mathbf{a}=(a_1,\dots ,a_k)$ of positive integers, let $V’_S($\mathbf{a}$)={ a_1s_1+\cdots+a_ks_k : s_i\in S} \setminus {0}$\mathrm{.}Forapositiveinteger$n$, let $\mathcal{T}(n)$ be the set of integers greater than or equal to $n$. In this paper, we consider the problem of finding all vectors $\mathbf{a}$ satisfying $V_S^{\mathrm{\prime }}(\mathbf{a})=\mathcal{T}(n)$, when $S$ is the set of (generalized) $m$-gonal numbers and $n$ is a positive integer. In particular, we completely resolve the case when $S$ is the set of triangular numbers.