On the nonvanishing of representation functions of some special sequences

For a given positive integer N, and any coloring function c : N -> {0, 1} satisfying c(2k) = 1 - c(k), c(2k + 1) = c(k) for all k >= N, we show that for all n >= 20N, n has both a monochromatic representation and a multicolored representation, in other words, there exist x, y, u, v is an element of N, such that n = x+y = u+v, c(x) = c(y) and c(u) not equal c(v). Similar results are obtained for another kind of coloring function c : N -> {0, 1} satisfying c(2k) = c(k) and c(2k + 1) = 1 - c(k) for all k >= N. This answers a question of Y.-G. Chen on the values of representation functions. (C) 2014 Elsevier, B.V. All rights reserved.