  For any integer $x$, let $T_x$ denote the triangular number
$\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all
the triples of positive integers $(\alpha, \beta, \gamma)$ for which the
ternary sums $\alpha x^2 +\beta T_y + \gamma T_z$ represent all but finitely
many positive integers. This resolves a conjecture of Kane and Sun
\cite[Conjecture 1.19(i)]{KS08} and complete the characterization of all almost
universal ternary mixed sums of squares and triangular numbers.
