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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Wythoff array</span></span>
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<p>In mathematics, the <b>Wythoff array</b> is an infinite <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> of <a href="Positive_integer" class="mw-redirect" title="Positive integer">positive integers</a> derived from the <a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> and named after Dutch mathematician <a href="Willem_Abraham_Wythoff" title="Willem Abraham Wythoff">Willem Abraham Wythoff</a>. 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.
</p><p>The Wythoff array was first defined by <a href="#CITEREFMorrison1980">Morrison (1980)</a> using Wythoff pairs, the coordinates of winning positions in <a href="Wythoff's_game" title="Wythoff's game">Wythoff's game</a>. It can also be defined using <a href="Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a> and <a href="Zeckendorf's_theorem" title="Zeckendorf's theorem">Zeckendorf's theorem</a>, or directly from the <a href="Golden_ratio" title="Golden ratio">golden ratio</a> and the <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a> defining the Fibonacci numbers.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Values">Values</h2></div>
<p>The Wythoff array has the values
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}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{matrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mn>13</mn>
</mtd>
<mtd>
<mn>21</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>11</mn>
</mtd>
<mtd>
<mn>18</mn>
</mtd>
<mtd>
<mn>29</mn>
</mtd>
<mtd>
<mn>47</mn>
</mtd>
<mtd>
<mn>76</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mn>10</mn>
</mtd>
<mtd>
<mn>16</mn>
</mtd>
<mtd>
<mn>26</mn>
</mtd>
<mtd>
<mn>42</mn>
</mtd>
<mtd>
<mn>68</mn>
</mtd>
<mtd>
<mn>110</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mn>15</mn>
</mtd>
<mtd>
<mn>24</mn>
</mtd>
<mtd>
<mn>39</mn>
</mtd>
<mtd>
<mn>63</mn>
</mtd>
<mtd>
<mn>102</mn>
</mtd>
<mtd>
<mn>165</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>12</mn>
</mtd>
<mtd>
<mn>20</mn>
</mtd>
<mtd>
<mn>32</mn>
</mtd>
<mtd>
<mn>52</mn>
</mtd>
<mtd>
<mn>84</mn>
</mtd>
<mtd>
<mn>136</mn>
</mtd>
<mtd>
<mn>220</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>14</mn>
</mtd>
<mtd>
<mn>23</mn>
</mtd>
<mtd>
<mn>37</mn>
</mtd>
<mtd>
<mn>60</mn>
</mtd>
<mtd>
<mn>97</mn>
</mtd>
<mtd>
<mn>157</mn>
</mtd>
<mtd>
<mn>254</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>17</mn>
</mtd>
<mtd>
<mn>28</mn>
</mtd>
<mtd>
<mn>45</mn>
</mtd>
<mtd>
<mn>73</mn>
</mtd>
<mtd>
<mn>118</mn>
</mtd>
<mtd>
<mn>191</mn>
</mtd>
<mtd>
<mn>309</mn>
</mtd>
<mtd>
<mo>⋯<!-- ⋯ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd>
<mo>⋱<!-- ⋱ --></mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}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{matrix}}}</annotation>
</semantics>
</math></span><img src="./5263b4f7334ad95ff33e904e349b9cc19695549b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:39.75ex; height:27.176ex;" alt="{\displaystyle {\begin{matrix}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{matrix}}}" loading="lazy"></span> (sequence <span class="nowrap external"><a href="https://oeis.org/A035513" class="extiw external" title="oeis:A035513">A035513</a></span> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Equivalent_definitions">Equivalent definitions</h2></div>
<p>Inspired by a similar Stolarsky array previously defined by <a href="#CITEREFStolarsky1977">Stolarsky (1977)</a>, <a href="#CITEREFMorrison1980">Morrison (1980)</a> defined the Wythoff array as follows. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mtext> </mtext>
<mspace width="negativethinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}</annotation>
</semantics>
</math></span><img src="./bf5ab0819fbf64ef2942ee3e805000ef40f24849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.701ex; height:3.509ex;" alt="{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}" loading="lazy"></span> denote the <a href="Golden_ratio" title="Golden ratio">golden ratio</a>; then the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>th winning position in <a href="Wythoff's_game" title="Wythoff's game">Wythoff's game</a> is given by the pair of positive integers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lfloor i\varphi \rfloor ,\lfloor i\varphi ^{2}\rfloor )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
<mi>i</mi>
<mi>φ<!-- φ --></mi>
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
<mo>,</mo>
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
<mi>i</mi>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\lfloor i\varphi \rfloor ,\lfloor i\varphi ^{2}\rfloor )}</annotation>
</semantics>
</math></span><img src="./df11931a519a1592a6c2fc7db5cf0274cad02217.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.672ex; height:3.176ex;" alt="{\displaystyle (\lfloor i\varphi \rfloor ,\lfloor i\varphi ^{2}\rfloor )}" loading="lazy"></span>, where the numbers on the left and right sides of the pair define two complementary <a href="Beatty_sequence" title="Beatty sequence">Beatty sequences</a> that together include each positive integer exactly once. Morrison defines the first two numbers in row <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
</semantics>
</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> of the array to be the Wythoff pair given by the equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=\lfloor m\varphi \rfloor }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
<mi>m</mi>
<mi>φ<!-- φ --></mi>
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i=\lfloor m\varphi \rfloor }</annotation>
</semantics>
</math></span><img src="./8df1d7d4de6c1c48bdaea241743c62ff9432634b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.526ex; height:2.843ex;" alt="{\displaystyle i=\lfloor m\varphi \rfloor }" loading="lazy"></span>, and where the remaining numbers in each row are determined by the Fibonacci recurrence relation. That is, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m,n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>,</mo>
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{m,n}}</annotation>
</semantics>
</math></span><img src="./f236c334e0b9516e4e9bc594461a5e3b120da767.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.862ex; height:2.843ex;" alt="{\displaystyle A_{m,n}}" loading="lazy"></span> denotes the entry in row <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
</semantics>
</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> and column <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> of the array, then
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m,1}=\left\lfloor \lfloor m\varphi \rfloor \varphi \right\rfloor }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
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<mi>m</mi>
<mo>,</mo>
<mn>1</mn>
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</msub>
<mo>=</mo>
<mrow>
<mo>⌊</mo>
<mrow>
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
<mi>m</mi>
<mi>φ<!-- φ --></mi>
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
<mi>φ<!-- φ --></mi>
</mrow>
<mo>⌋</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{m,1}=\left\lfloor \lfloor m\varphi \rfloor \varphi \right\rfloor }</annotation>
</semantics>
</math></span><img src="./3cca11217f844a7bd94f4c066e06acccc77856c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.006ex; height:3.009ex;" alt="{\displaystyle A_{m,1}=\left\lfloor \lfloor m\varphi \rfloor \varphi \right\rfloor }" loading="lazy"></span>,</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m,2}=\left\lfloor \lfloor m\varphi \rfloor \varphi ^{2}\right\rfloor }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>,</mo>
<mn>2</mn>
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</msub>
<mo>=</mo>
<mrow>
<mo>⌊</mo>
<mrow>
<mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo>
<mi>m</mi>
<mi>φ<!-- φ --></mi>
<mo fence="false" stretchy="false">⌋<!-- ⌋ --></mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
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<mo>⌋</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{m,2}=\left\lfloor \lfloor m\varphi \rfloor \varphi ^{2}\right\rfloor }</annotation>
</semantics>
</math></span><img src="./892ecc76203d567585f792e96b2dbfede7b3ad33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.19ex; height:3.343ex;" alt="{\displaystyle A_{m,2}=\left\lfloor \lfloor m\varphi \rfloor \varphi ^{2}\right\rfloor }" loading="lazy"></span>, and</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m,n}=A_{m,n-2}+A_{m,n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>,</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
<mo>,</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
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<mo>+</mo>
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<mi>A</mi>
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<mi>m</mi>
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<mo>−<!-- − --></mo>
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<annotation encoding="application/x-tex">{\displaystyle A_{m,n}=A_{m,n-2}+A_{m,n-1}}</annotation>
</semantics>
</math></span><img src="./fc8f71a8d6e26475e018ebe0c63c5ccef4fcd492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.725ex; height:2.843ex;" alt="{\displaystyle A_{m,n}=A_{m,n-2}+A_{m,n-1}}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>2}">
<semantics>
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle n>2}</annotation>
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</math></span><img src="./44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>2}" loading="lazy"></span>.</dd></dl>
<p>The <a href="Zeckendorf's_theorem" title="Zeckendorf's theorem">Zeckendorf representation</a> 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 <a href="#CITEREFKimberling1995">Kimberling (1995)</a> 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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{m,n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
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<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle A_{m,n}}</annotation>
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</math></span><img src="./f236c334e0b9516e4e9bc594461a5e3b120da767.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.862ex; height:2.843ex;" alt="{\displaystyle A_{m,n}}" loading="lazy"></span> of the array is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">
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<mi>m</mi>
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</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span>th smallest number whose Zeckendorf representation begins with the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>n</mi>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle (n+1)}</annotation>
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</math></span><img src="./b30a29cfd35628469f9dbffea4804f5b422f3037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n+1)}" loading="lazy"></span>th Fibonacci number.
</p>
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<p>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 (<a href="#CITEREFMorrison1980">Morrison 1980</a>).
</p><p>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 <a href="Lucas_number" title="Lucas number">Lucas numbers</a> appears in shifted form in the second row (<a href="#CITEREFMorrison1980">Morrison 1980</a>).
</p>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFKimberling1995" class="citation cs2"><a href="Clark_Kimberling" title="Clark Kimberling">Kimberling, Clark</a> (1995), <a rel="nofollow" class="external text" href="http://www.fq.math.ca/Scanned/33-1/kimberling.pdf">"The Zeckendorf array equals the Wythoff array"</a> <span class="cs1-format">(PDF)</span>, <i><a href="Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></i>, <b>33</b> (1): <span class="nowrap">3–</span>8, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00150517.1995.12429166">10.1080/00150517.1995.12429166</a></cite>.</li>
<li><cite id="CITEREFMorrison1980" class="citation cs2"><a href="David_R._Morrison_(mathematician)" title="David R. Morrison (mathematician)">Morrison, D. R.</a> (1980), "A Stolarsky array of Wythoff pairs", <a rel="nofollow" class="external text" href="http://web.math.ucsb.edu/~drm/papers/stolarsky.pdf"><i>A Collection of Manuscripts Related to the Fibonacci Sequence</i></a> <span class="cs1-format">(PDF)</span>, Santa Clara, Calif: The Fibonacci Association, pp. <span class="nowrap">134–</span>136</cite>.</li>
<li><cite id="CITEREFStolarsky1977" class="citation cs2">Stolarsky, K. B. (1977), <a rel="nofollow" class="external text" href="http://www.fq.math.ca/Scanned/15-3/stolarsky.pdf">"A set of generalized Fibonacci sequences such that each natural number belongs to exactly one"</a> <span class="cs1-format">(PDF)</span>, <i><a href="Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></i>, <b>15</b> (3): 224, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00150517.1977.12430440">10.1080/00150517.1977.12430440</a></cite>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><span class="citation mathworld" id="Reference-Mathworld-Wythoff_Array"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/WythoffArray.html">"Wythoff Array"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>
<li><a rel="nofollow" class="external text" href="https://oeis.org/A035513/">The Wythoff Array (Online Encyclopedia of Integer Sequences)</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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