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Let $P_8(x)=3x^2-2x$. For positive integers $a_1,a_2,\dots,a_k$, a polynomialof the form $a_1P_8(x_1)+a_2P_8(x_2)+\cdots+a_kP_8(x_k)$ is called an octagonalform. For a positive integer $n$, an octagonal form is called tight $\mathcalT(n)$-universal if it represents (over $\mathbb{z}$) every positive integergreater than or equal to $n$ and does not represent any positive integer lessthan $n$. In this article, we find all tight $\mathcal T(n)$-universaloctagonal forms for every $n\ge 2$. Furthermore, we provide an effectivecriterion on tight $\mathcal T(n)$-universality of an arbirary octagonal form,which is a generalization of "15-Theorem" of Conway and Schneeberger.

Tight universal octagonal forms