In mathematics, the Wythoff array is an infinite matrix of positive integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff.
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.
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.
, 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.
, and where the remaining numbers in each row are determined by the Fibonacci recurrence relation.
of the array, then The Zeckendorf representation of any positive integer is a representation as a sum of distinct Fibonacci numbers, no two of which are consecutive in the Fibonacci sequence.
th smallest number whose Zeckendorf representation begins with the
Each Wythoff pair occurs exactly once in the Wythoff array, as a consecutive pair of numbers in the same row, with an odd index for the first number and an even index for the second.
Every sequence of positive integers satisfying the Fibonacci recurrence occurs, shifted by at most finitely many positions, in the Wythoff array.
In particular, the Fibonacci sequence itself is the first row, and the sequence of Lucas numbers appears in shifted form in the second row (Morrison 1980).