Wythoff array: Difference between revisions

Revision as of 20:52, 7 March 2016

Revision as of 13:02, 3 August 2017

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==Equivalent definitions==

Inspired by a similar array previously defined by {{harvtxt|Stolarsky|1977}}, {{harvtxt|Morrison|1980}} defined the Wythoff array as follows. Let <math>\varphi=\frac{1+\sqrt{5}}{2}</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