The Wythoff difference array (Q2883398)

The Wythoff game is a two-player game where, given a pair of positive integers, the players take turns choosing a positive integer to subtract from one or both terms. The player who gets both integers to zero wins. A Wythoff pair is a pair of positive integers such that if a player can end their turn on a Wythoff pair, then they can always reach a Wythoff pair from then on and win the game, regardless of the other player's moves. If we list the Wythoff pairs with the smaller term first, we obtain a pair of sequences which partition the natural numbers. Moreover, these sequences can be generated by applying the floor function to positive integer multiples of the golden ratio and its square. Indeed, we can arrange the Wythoff pairs into a remarkable two dimensional array \(W\), the Wythoff array, whose rows are exactly the generalized Fibonacci sequences, up to a shift. The author defines the Wythoff difference array \(D\), which is obtained by subtracting the odd-numbered columns from the even-numbered columns of \(W\), and proves that \(D\) has various interesting combinatorial and fractal properties related to the Fibonacci numbers and the Wythoff array.