Wythoff array: Difference between revisions

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{{short description|Infinite matrix of integers derived from the Fibonacci sequence}}

In mathematics, the '''Wythoff array''' is an infinite [[Matrix (mathematics)|matrix]] of [[positive integer]]s derived from the [[Fibonacci sequence]] and named after Dutch mathematician [[Willem Abraham Wythoff]]. It was first defined by {{harvtxt|Morrison|1980}} using Wythoff pairs, the coordinates of winning positions in [[Wythoff's game]]; it can also be defined using [[Fibonacci number]]s 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 was first defined by {{harvtxt|Morrison|1980}} using Wythoff pairs, the coordinates of winning positions in [[Wythoff's game]]. It can also be defined using [[Fibonacci number]]s and [[Zeckendorf's theorem]], or directly from the [[golden ratio]] and the [[recurrence relation]] defining the Fibonacci numbers.

==Values==

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==Equivalent definitions==

Inspired by a similar [[Stolarsky array]] previously defined by {{harvtxt|Stolarsky|1977}}, {{harvtxt|Morrison|1980}} defined the Wythoff array as follows. Let <math>\varphi=\frac{tfrac12\bigl(1+\sqrt{5}}{2}~\!\bigr)</math> denote the [[golden ratio]]; then the <math>i</math>th winning position in [[Wythoff's game]] is given by the pair of positive integers <math>(\lfloor i\varphi\rfloor, \lfloor i\varphi^2\rfloor)</math>, where the numbers on the left and right sides of the pair define two complementary [[Beatty sequence]]s that together include each positive integer exactly once. Morrison defines the first two numbers in row <math>m</math> of the array to be the Wythoff pair given by the equation <math>mi=\lfloor im\varphi\rfloor</math>, and where the remaining numbers in each row are determined by the Fibonacci recurrence relation. That is, if <math>A_{m,n}</math> denotes the entry in row <math>m</math> and column <math>n</math> of the array, then

:<math>A_{m,1} = \left\lfloor \lfloor m\varphi \rfloor \varphi \right\rfloor</math>,

:<math>A_{m,2} = \left\lfloor \lfloor m\varphi \rfloor \varphi^2 \right\rfloor</math>, and

:<math>A_{m,n} = A_{m,n-2}+A_{m,n-1}</math> for <math>n > 2</math>.

The [[Zeckendorf's theorem|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 {{harvtxt|Kimberling|1995}} 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 <math>A_{m,n}</math> of the array is the <math>m</math>th smallest number whose Zeckendorf representation begins with the <math>(n+1)</math>th Fibonacci number.

==Properties==

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| url = http://www.fq.math.ca/Scanned/33-1/kimberling.pdf

| volume = 33

| year = 1995| doi = 10.1080/00150517.1995.12429166 }}.

*{{citation

| last = Morrison | first = D. R. | author-link = David R. Morrison (mathematician)

| contribution = A Stolarsky array of Wythoff pairs

| ___location = Santa Clara, Calif

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| publisher = The Fibonacci Association

| title = A Collection of Manuscripts Related to the Fibonacci Sequence

| url = http://wwwweb.math.ucsb.edu/~drm/papers/stolarsky.pdf

| year = 1980}}.

*{{citation

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| url = http://www.fq.math.ca/Scanned/15-3/stolarsky.pdf

| volume = 15

| year = 1977}}| doi = 10.1080/00150517.1977.12430440

}}.

==External links==

*{{mathworld|urlname=WythoffArray|title=Wythoff Array}}

*[https://oeis.org/A035513/ The Wythoff Array (Online Encyclopedia of Integer Sequences)]

[[Category:Triangles of numbers]]