Wythoff array

In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. It was first defined by Morrison (1980) using Wythoff pairs, the coordinates of winning positions in Wythoff's game; it can also be defined using Fibonacci numbers and Zeckendorf's theorem, or directly from the golden ratio and the recurrence relation defining the Fibonacci numbers. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.

The Wythoff array has the values

Inspired by a similar array previously defined by Stolarsky (1977), Morrison (1980) defined the Wythoff array as follows. Let ${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$ denote the golden ratio; then the ${\displaystyle i}$th winning position in Wythoff's game is given by the pair of positive integers ${\displaystyle (\lfloor i\varphi \rfloor ,\lfloor i\varphi ^{2}\rfloor )}$, where the numbers on the left and right sides of the pair define two complementary Beatty sequences that together include each positive integer exactly once. Morrison defines the first two numbers in row ${\displaystyle m}$ of the array to be the Wythoff pair given by the equation ${\displaystyle m=\lfloor i\varphi \rfloor }$, and where the remaining numbers in each row are determined by the Fibonacci recurrence relation. That is, if ${\displaystyle A_{m,n}}$ denotes the entry in row ${\displaystyle m}$ and column ${\displaystyle n}$ of the array, then

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