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If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ 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 $\\mathbb Z$. 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If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ 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 $\\mathbb Z$. 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