  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
$\Delta_{\alpha_1,\alpha_2,\dots,\alpha_k}(x_1,x_2,\dots,x_k)=n$ 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.
