Some extensions of a property of linear representation functions

Let A = {a(1), a(2), ...}(a(1) < a(2) < ...) be an infinite sequence of nonnegative integers. Let k >= 2 be a fixed integer and for n is an element of N, let R-k(A, n) be the number of solutions of a(i1) + ... + a(ik) = n, a(i1), ..., a(ik) is an element of A, and let R-k((1)) (A, n) and R-k((2))(A, n) denote the number of solutions with the additional restrictions a(i1) < ... < a(ik), and a(i1) <= ... <= a(ik) respectively. Recently, Horvath proved that if d > 0 is an integer, then there does not exist n(0) such that d <= R-2((2)) (A, n) <= d + [root 2d + 1/2] for n > n(0). In this paper, we obtain the analogous results for R-k(A, n), R-k((1))(A, n) and R-k((2)) (A, n). (C) 2009 Elsevier B.V. All rights reserved.