let $0 < alpha < beta$ be two irrational numbers satisfying $1/alpha + 1/beta = 1$. then the sequences $a'_n = lfloor {nalpha}rfloor$, $b'_n = lfloor{nbeta}rfloor$, $nge 1$, are complementary over $iz_{ge 1}$. thus $a'_n = {rm mex_1} {a'_i,b'_i : 1 le i< n}$, $n geq 1$ (${rm mex_1}(s)$, the smallest positive integer not in the set $s$). suppose that $c = beta-alpha$ is an integer. then $b'_n = a'_n+cn$ for all $n ge 1$. we define the following generalization of the sequences $a'_n$, $b'_n$: let $c,;n_0iniz_{ge 1}$, and let $xsubsetiz_{ge 1}$ be an arbitrary finite set. let $a_n = {rm mex_1}(xcup{a_i,b_i : 1 leq i< n})$, $b_n = a_n+cn$, $nge n_0$. let $s_n = a'_n-a_n$. we show that no matter how we pick $c,;n_0$ and $x$, from some point on the {it shift sequence/} $s_n$ assumes either one constant value or three successive values; and if the second case holds, it assumes these values in a very distinct fractal-like pattern, which we describe. this work was motivated by a generalization of wythoff's game to $nge 3$ piles.