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

In this paper we first investigate for what positive integers a,b,c every nonnegative integer n can be written 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⩽b⩽c⩽d⩽a with a>2, we prove that any nonnegative integer can be written 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).