Queen reflections: a modification of Wythoff Nim

Wythoff Nim is a classical combinatorial game of queen moves on a chessboard. There are many ways to describe its P-positions (safe positions to move to). One way is to code them by the Fibonacci word 010010100100101.., which is the unique fixed point of the substitution of 0 by 01, and of 1 by 0. The coordinates of the n-th P-position are...