In mathematics, the Wythoff array is an infinite

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s derived from the

Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

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. The Wythoff array was first defined by using Wythoff pairs, the coordinates of winning positions in Wythoff's game. It can also be defined using

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

s and

Zeckendorf's theorem In mathematics, Zeckendorf's theorem, named after Belgian amateur mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers. Zeckendorf's theorem states that every positive integer can be r ...

, or directly from the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

and the

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

defining the Fibonacci numbers.

Values

The Wythoff array has the values :

\begin
1&2&3&5&8&13&21&\cdots\\
4&7&11&18&29&47&76&\cdots\\
6&10&16&26&42&68&110&\cdots\\
9&15&24&39&63&102&165&\cdots\\
12&20&32&52&84&136&220&\cdots\\
14&23&37&60&97&157&254&\cdots\\
17&28&45&73&118&191&309&\cdots\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\
\end

Equivalent definitions

Inspired by a similar Stolarsky array previously defined by , defined the Wythoff array as follows. Let

\varphi=\frac

denote the

; then the

i

th winning position in Wythoff's game is given by the pair of positive integers

(\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 sequence In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about th ...

s that together include each positive integer exactly once. Morrison defines the first two numbers in row

m

of the array to be the Wythoff pair given by the equation

i=\lfloor m\varphi\rfloor

, and where the remaining numbers in each row are determined by the Fibonacci recurrence relation. That is, if

A_

denotes the entry in row

m

and column

n

of the array, then :

A_ = \left\lfloor \lfloor m\varphi \rfloor \varphi \right\rfloor

, :

A_ = \left\lfloor \lfloor m\varphi \rfloor \varphi^2 \right\rfloor

, and :

A_ = A_+A_

for

n > 2

. 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. As describes, the numbers within each row of the array have Zeckendorf representation that differ by a shift operation from each other, and the numbers within each column have Zeckendorf representations that all use the same smallest Fibonacci number. In particular the entry

A_

of the array is the

m

th smallest number whose Zeckendorf representation begins with the

(n+1)

th Fibonacci number.

Properties

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. Because each positive integer occurs in exactly one Wythoff pair, each positive integer occurs exactly once in the array . 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 number The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci n ...

s appears in shifted form in the second row .

References

*. *. *.

External links

*{{mathworld, urlname=WythoffArray, title=Wythoff Array Triangles of numbers Fibonacci numbers