Tight universal octagonal forms

Let \(P_8(x)=3x^2-2x\) $P_8(x)=3x^2-2x$ . For positive integers \(a_1,a_2,\dots ,a_k\) $a_1,a_2,\cdots ,a_k$ , a polynomial of the form \(a_1P_8(x_1)+a_2P_8(x_2)+\cdots +a_kP_8(x_k)\) $a_1{P}_8(x_1)+a_2{P}_8(x_2)+\cdots +a_k{P}_8(x_k)$ is called an octagonal form. For a positive integer n, an octagonal form is called tight \(\mathcal T(n)\) $\mathcal{T}(n)$ -universal if it represents (over \({\mathbb Z}\) $\mathbb{Z}$ ) every positive integer greater than or equal to n and does not represent any positive integer less than n. In this article, we find all tight \(\mathcal T(n)\) $\mathcal{T}(n)$ -universal octagonal forms for every \(n\ge 2\) $n\ge 2$ .