On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$

Abstract: In this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. We show that $(a,b,c)$ can be either of the following seven triples: $$(1,2,3),\ (1,2,4),\ (1,2,5),\ (2,2,4),\ (2,2,5),\ (2,3,3),\ (2,3,4),$$ and conjecture that any triple $(a,b,c)$ among $$(2,2,6),\ (2,3,5),\ (2,3,7),\ (2,3,8),\ (2,3,9),\ (2,3,10)$$ also has the desired property. For integers $0\le b\le c\le d\le a$ with $a>2$, we prove that any nonnegative integer can be represented as $x(ax+b)+y(ay+c)+z(az+d)$ with $x,y,z$ integers, if and only if the quadruple $(a,b,c,d)$ is among $$(3,0,1,2),\ (3,1,1,2),\ (3,1,2,2),\ (3,1,2,3),\ (4,1,2,3).$$