Some universal quadratic sums over the integers

Let $a,b,c,d,e,f\in \mathbb{N}$ with $a\ge c\ge e>0$, $b\le a$ and $b\equiv a\pmod2$,$d\le c$and$d\equiv c\pmod2$,$f\le e$and$f\equiv e\pmod2$. If any nonnegative integer can be written as $x(ax+b)\mathrm{/}2+y(cy+d)\mathrm{/}2+z(ez+f)\mathrm{/}2$ with $x,y,z\in \mathbb{Z}$, then the ordered tuple $(a,b,c,d,e,f)$ is said to be universal over $\mathbb{Z}$. Recently, Z.-W. Sun found all candidates for such universal tuples over $\mathbb{Z}$. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples $(a,b,c,d,e,f)$ in Sun’s list of candidates are indeed universal over $\mathbb{Z}$. For example, we prove the universality of $(16,4,2,0,1,1)$ over $\mathbb{Z}$ which is related to the form $x^2+y^2+32z^2$.