Almost universal sums of triangular numbers with one exception

For an arbitrary integer $x$, an integer of the form $T(x)=\frac{x^2+x}{2}$ is called a triangular number. For positive integers ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k$, a sum $\Delta_{\alpha_1,\alpha_2,\dots,\alpha_k}(x_1,x_2,\dots,x_k)=\alpha_1 T(x_1)+\alpha_2 T(x_2)+\cdots+\alpha_k T(x_k)$ of triangular numbers is said to be almost universal with one exception if the Diophantine equation ${\mathrm{\Delta }}_{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k}(x_1,x_2,\dots ,x_k)=n$ has an integer solution $(x_1,x_2,\dots ,x_k)\in {\mathbb{Z}}^k$ for any nonnegative integer $n$ except a single one. In this article, we classify all almost universal sums of triangular numbers with one exception. Furthermore, we provide an effective criterion on almost universality with one exception of an arbitrary sum of triangular numbers, which is a generalization of “15-theorem” of Conway, Miller and Schneeberger.