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∈N, such that n=x+y=u+v, c(x)=c(y) and c(u)≠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.