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 ∈ N , let R k ( A , n ) be the number of solutions of a i 1 + ⋯ + a i k = n , a i 1 , … , a i k ∈ A , and let R k ( 1 ) ( A , n ) and R k ( 2 ) ( A , n ) denote the number of solutions with the additional restrictions a i 1 < ⋯ < a i k , and a i 1 ≤ ⋯ ≤ a i k respectively. Recently, Horváth proved that if d > 0 is an integer, then there does not exist n 0 such that d ≤ R 2 ( 2 ) ( A , n ) ≤ d + [ 2 d + 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 ) .