Some extensions of a property of linear representation functions

AbstractLet A={a1,a2,…}(a1<a2<⋯) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for n∈N, let Rk(A,n) be the number of solutions of ai1+⋯+aik=n,ai1,…,aik∈A, and let Rk(1)(A,n) and Rk(2)(A,n) denote the number of solutions with the additional restrictions ai1<⋯<aik, and ai1≤⋯≤aik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n0 such that d≤R2(2)(A,n)≤d+[2d+12] for n>n0. In this paper, we obtain the analogous results for Rk(A,n), Rk(1)(A,n) and Rk(2)(A,n)