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Baum–Sweet sequence - Wikipedia

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free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Baum-Sweet_sequence&amp;redirect=no" class="mw-redirect" title="Baum-Sweet sequence">Baum-Sweet sequence</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical sequence of 1s and 0s</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> the <b>Baum–Sweet sequence</b> is an infinite <a href="/wiki/Automatic_sequence" title="Automatic sequence">automatic sequence</a> of 0s and 1s defined by the rule: </p> <dl><dd><i>b</i><sub><i>n</i></sub> = 1 if the binary representation of <i>n</i> contains no block of consecutive 0s of odd length;</dd> <dd><i>b</i><sub><i>n</i></sub> = 0 otherwise;</dd></dl> <p>for <i>n</i> ≥ 0.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup> </p><p>For example, <i>b</i><sub>4</sub> = 1 because the binary representation of 4 is 100, which only contains one block of consecutive 0s of length 2; whereas <i>b</i><sub>5</sub> = 0 because the binary representation of 5 is 101, which contains a block of consecutive 0s of length 1. </p><p>Starting at <i>n</i> = 0, the first few terms of the Baum–Sweet sequence are: </p> <dl><dd>1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1 ... (sequence <span class="nowrap external"><a href="//oeis.org/A086747" class="extiw" title="oeis:A086747">A086747</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historical_motivation">Historical motivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Baum%E2%80%93Sweet_sequence&amp;action=edit&amp;section=1" title="Edit section: Historical motivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The properties of the sequence were first studied by <a href="/wiki/Leonard_E._Baum" title="Leonard E. Baum">Leonard E. Baum</a> and Melvin M. Sweet in 1976.<sup id="cite_ref-BaumSweet_2-0" class="reference"><a href="#cite_note-BaumSweet-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup> In 1949, <a href="/wiki/Aleksandr_Khinchin" title="Aleksandr Khinchin">Khinchin</a> conjectured that there does not exist a non-quadratic algebraic real number having bounded partial quotients in its continued fraction expansion. A counterexample to this conjecture is still not known.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup> Baum and Sweet's paper showed that the same expectation is not met for algebraic <a href="/wiki/Power_series" title="Power series">power series</a>. They gave an example of cubic power series in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}((X^{-1}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}((X^{-1}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33bcc597de50bf125160f083649bdb5818b3769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.423ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{2}((X^{-1}))}"></span> whose partial quotients are bounded. (The degree of the power series in Baum and Sweet's result is analogous to the degree of the field extension associated with the algebraic real in Khinchin's conjecture.) </p><p>One of the series considered in Baum and Sweet's paper is a root of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{3}+x^{-1}f+1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{3}+x^{-1}f+1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34b5322979cc0460487e7b5ca54e950965251d7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.067ex; height:3.009ex;" alt="{\displaystyle f^{3}+x^{-1}f+1=0.}"></span><sup id="cite_ref-BaumSweet_2-1" class="reference"><a href="#cite_note-BaumSweet-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">&#91;</span>5<span class="cite-bracket">&#93;</span></a></sup></dd></dl> <p>The authors show that by <a href="/wiki/Hensel%27s_lemma" title="Hensel&#39;s lemma">Hensel's lemma</a>, there is a unique such root in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{2}((X^{-1}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{2}((X^{-1}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33bcc597de50bf125160f083649bdb5818b3769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.423ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{2}((X^{-1}))}"></span> because reducing the defining equation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> modulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f75a18781a402f9fb09972f99e092bf21aa877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.33ex; height:2.676ex;" alt="{\displaystyle X^{-1}}"></span> gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{3}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{3}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a437054df655030df7992c43600b15f492a4a390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.378ex; height:3.009ex;" alt="{\displaystyle f^{3}+1}"></span>, which factors as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{3}+1=(f+1)(f^{2}+f+1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>f</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{3}+1=(f+1)(f^{2}+f+1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e3978ea442731127ced5bf6cdb8c57a4635751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.52ex; height:3.176ex;" alt="{\displaystyle f^{3}+1=(f+1)(f^{2}+f+1).}"></span></dd></dl> <p>They go on to prove that this unique root has partial quotients of degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>&#x2264;<!-- ≤ --></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b666db35b341e1f27eafd54b901db4bbe7e9c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.616ex; height:2.343ex;" alt="{\displaystyle \leq 2}"></span>. Before doing so, they state (in the remark following Theorem 2, p 598)<sup id="cite_ref-BaumSweet_2-2" class="reference"><a href="#cite_note-BaumSweet-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup> that the root can be written in the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{k\geq 0}f_{i}X^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>&#x2265;<!-- ≥ --></mo> <mn>0</mn> </mrow> </munder> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2212;<!-- − --></mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{k\geq 0}f_{i}X^{-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891b91996f5c08368f6a317d55621158210d186f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.422ex; height:5.843ex;" alt="{\displaystyle f=\sum _{k\geq 0}f_{i}X^{-k}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a03ee597bc6f91db290f07795f38684e5e7f9805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.454ex; height:2.509ex;" alt="{\displaystyle f_{0}=1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{k}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{k}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79850a4b486256c3fa969b6bf70c8c3320b9db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.489ex; height:2.509ex;" alt="{\displaystyle f_{k}=1}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>&#x2265;<!-- ≥ --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}"></span> if and only if the binary expansion of <span class="mwe-math-element"><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> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> contains only even length blocks of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>'s. This is the origin of the Baum–Sweet sequence. </p><p>Mkaouar<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">&#91;</span>6<span class="cite-bracket">&#93;</span></a></sup> and Yao<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">&#91;</span>7<span class="cite-bracket">&#93;</span></a></sup> proved that the partial quotients of the continued fraction for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> above do not form an automatic sequence.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">&#91;</span>8<span class="cite-bracket">&#93;</span></a></sup> However, the sequence of partial quotients can be generated by a non-uniform morphism.<sup id="cite_ref-FinAut_9-0" class="reference"><a href="#cite_note-FinAut-9"><span class="cite-bracket">&#91;</span>9<span class="cite-bracket">&#93;</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Baum%E2%80%93Sweet_sequence&amp;action=edit&amp;section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Baum–Sweet sequence can be generated by a 3-state <a href="/wiki/Finite_state_machine" class="mw-redirect" title="Finite state machine">automaton</a>.<sup id="cite_ref-FinAut_9-1" class="reference"><a href="#cite_note-FinAut-9"><span class="cite-bracket">&#91;</span>9<span class="cite-bracket">&#93;</span></a></sup> </p><p>The value of term <i>b</i><sub><i>n</i></sub> in the Baum–Sweet sequence can be found recursively as follows. If <i>n</i> = <i>m</i>·4<sup><i>k</i></sup>, where <i>m</i> is not divisible by 4 (or is 0), then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}={\begin{cases}1&amp;{\text{if }}n=0\\0&amp;{\text{if }}m{\text{ is even}}\\b_{(m-1)/2}&amp;{\text{if }}m{\text{ is odd}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if&#xA0;</mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if&#xA0;</mtext> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>&#xA0;is even</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if&#xA0;</mtext> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>&#xA0;is odd</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}={\begin{cases}1&amp;{\text{if }}n=0\\0&amp;{\text{if }}m{\text{ is even}}\\b_{(m-1)/2}&amp;{\text{if }}m{\text{ is odd}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd3a81d3640da52e80897910a4af8cc0952a620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.558ex; margin-bottom: -0.28ex; width:29.44ex; height:8.843ex;" alt="{\displaystyle b_{n}={\begin{cases}1&amp;{\text{if }}n=0\\0&amp;{\text{if }}m{\text{ is even}}\\b_{(m-1)/2}&amp;{\text{if }}m{\text{ is odd}}.\end{cases}}}"></span></dd></dl> <p>Thus b<sub>76</sub> = <i>b</i><sub>9</sub> = <i>b</i><sub>4</sub> = <i>b</i><sub>0</sub> = 1, which can be verified by observing that the binary representation of 76, which is 1001100, contains no consecutive blocks of 0s with odd length. </p><p>The Baum–Sweet word 1101100101001001..., which is created by concatenating the terms of the Baum–Sweet sequence, is a fixed point of the morphism or <a href="/wiki/String_substitution" class="mw-redirect" title="String substitution">string substitution</a> rules </p> <dl><dd>00 <b>→</b> 0000</dd> <dd>01 <b>→</b> 1001</dd> <dd>10 <b>→</b> 0100</dd> <dd>11 <b>→</b> 1101</dd></dl> <p>as follows: </p> <dl><dd>11 <b>→</b> 1101 <b>→</b> 11011001 <b>→</b> 1101100101001001 <b>→</b> 11011001010010011001000001001001 ...</dd></dl> <p>From the morphism rules it can be seen that the Baum–Sweet word contains blocks of consecutive 0s of any length (<i>b</i><sub><i>n</i></sub>&#160;=&#160;0 for all 2<sup><i>k</i></sup> integers in the range 5.2<sup><i>k</i></sup> &#8804; <i>n</i>&#160;&lt;&#160;6.2<sup><i>k</i></sup>), but it contains no block of three consecutive 1s. </p><p>More succinctly, by <a href="/wiki/Automatic_sequence#Substitution" title="Automatic sequence">Cobham's little theorem</a> the Baum–Sweet word can be expressed as a coding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03C4;<!-- τ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> applied to the fixed point of a uniform morphism <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03C6;<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>. Indeed, the morphism </p> <dl><dd><span class="mwe-math-element"><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 (A)=AB,\varphi (B)=CB,\varphi (C)=BD,\varphi (D)=DD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03C6;<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>B</mi> <mo>,</mo> <mi>&#x03C6;<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mi>B</mi> <mo>,</mo> <mi>&#x03C6;<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>B</mi> <mi>D</mi> <mo>,</mo> <mi>&#x03C6;<!-- φ --></mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>D</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (A)=AB,\varphi (B)=CB,\varphi (C)=BD,\varphi (D)=DD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a427181b0e79a431e4713b771fc8c866b19683a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.585ex; height:2.843ex;" alt="{\displaystyle \varphi (A)=AB,\varphi (B)=CB,\varphi (C)=BD,\varphi (D)=DD}"></span></dd></dl> <p>and coding </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau (A)=1,\tau (B)=1,\tau (C)=0,\tau (D)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03C4;<!-- τ --></mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>&#x03C4;<!-- τ --></mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>&#x03C4;<!-- τ --></mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>&#x03C4;<!-- τ --></mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (A)=1,\tau (B)=1,\tau (C)=0,\tau (D)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa1c025fdd463a3c6ff686f6eecaa07df3b7542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.388ex; height:2.843ex;" alt="{\displaystyle \tau (A)=1,\tau (B)=1,\tau (C)=0,\tau (D)=0}"></span></dd></dl> <p>generate the word in that way.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">&#91;</span>10<span class="cite-bracket">&#93;</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Baum%E2%80%93Sweet_sequence&amp;action=edit&amp;section=3" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Baum–Sweet_Sequence"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Baum-SweetSequence.html">"Baum–Sweet Sequence"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=MathWorld&amp;rft.atitle=Baum%E2%80%93Sweet+Sequence&amp;rft.au=Weisstein%2C+Eric+W.&amp;rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FBaum-SweetSequence.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABaum%E2%80%93Sweet+sequence" class="Z3988"></span></span></span> </li> <li id="cite_note-BaumSweet-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-BaumSweet_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BaumSweet_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-BaumSweet_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaumSweet1976" class="citation journal cs1">Baum, Leonard E.; Sweet, Melvin M. (1976). "Continued Fractions of Algebraic Power Series in Characteristic 2". <i>Annals of Mathematics</i>. <b>103</b> (3): 593–610. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970953">10.2307/1970953</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970953">1970953</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Annals+of+Mathematics&amp;rft.atitle=Continued+Fractions+of+Algebraic+Power+Series+in+Characteristic+2&amp;rft.volume=103&amp;rft.issue=3&amp;rft.pages=593-610&amp;rft.date=1976&amp;rft_id=info%3Adoi%2F10.2307%2F1970953&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970953%23id-name%3DJSTOR&amp;rft.aulast=Baum&amp;rft.aufirst=Leonard+E.&amp;rft.au=Sweet%2C+Melvin+M.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABaum%E2%80%93Sweet+sequence" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWaldschmidt2009" class="citation book cs1">Waldschmidt, M. (2009). "Words and Transcendence". In W.W.L. Chen; W.T. Gowers; H. Halbertstam; W.M. Schmidt; R.C. Vaughan (eds.). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-00407221/file/WordsTranscendence.pdf"><i>Analytic Number Theory: Essays in Honour of Klaus Roth</i></a> <span class="cs1-format">(PDF)</span>. Cambridge University Press. Section 31, p.&#160;449–470.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Words+and+Transcendence&amp;rft.btitle=Analytic+Number+Theory%3A+Essays+in+Honour+of+Klaus+Roth&amp;rft.pages=Section+31%2C+p.-449-470&amp;rft.pub=Cambridge+University+Press&amp;rft.date=2009&amp;rft.aulast=Waldschmidt&amp;rft.aufirst=M.&amp;rft_id=https%3A%2F%2Fhal.archives-ouvertes.fr%2Fhal-00407221%2Ffile%2FWordsTranscendence.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABaum%E2%80%93Sweet+sequence" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhinchin1964" class="citation book cs1">Khinchin, A.I. (1964). <i>Continued Fractions</i>. University of Chicago Press.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Continued+Fractions&amp;rft.pub=University+of+Chicago+Press&amp;rft.date=1964&amp;rft.aulast=Khinchin&amp;rft.aufirst=A.I.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABaum%E2%80%93Sweet+sequence" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Graham Everest, Alf van der Poorten, Igor Shparlinski, Thomas Ward <i>Recurrence Sequences</i> AMS 2003, p&#160;236.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMkaouar1995" class="citation journal cs1">Mkaouar, M. (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.2264">"Sur le développement en fraction continue de la série de Baum et Sweet"</a>. <i>Bull. Soc. Math. France</i>. <b>123</b> (3): 361–374. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.2264">10.24033/bsmf.2264</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Bull.+Soc.+Math.+France&amp;rft.atitle=Sur+le+d%C3%A9veloppement+en+fraction+continue+de+la+s%C3%A9rie+de+Baum+et+Sweet.&amp;rft.volume=123&amp;rft.issue=3&amp;rft.pages=361-374&amp;rft.date=1995&amp;rft_id=info%3Adoi%2F10.24033%2Fbsmf.2264&amp;rft.aulast=Mkaouar&amp;rft.aufirst=M.&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.24033%252Fbsmf.2264&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABaum%E2%80%93Sweet+sequence" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYao1997" class="citation journal cs1">Yao, J.-Y. (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa-80-3-237-248">"Critères de non-automaticité et leurs applications"</a>. <i>Acta Arith</i>. <b>80</b> (3): 237–248. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Faa-80-3-237-248">10.4064/aa-80-3-237-248</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Acta+Arith.&amp;rft.atitle=Crit%C3%A8res+de+non-automaticit%C3%A9+et+leurs+applications.&amp;rft.volume=80&amp;rft.issue=3&amp;rft.pages=237-248&amp;rft.date=1997&amp;rft_id=info%3Adoi%2F10.4064%2Faa-80-3-237-248&amp;rft.aulast=Yao&amp;rft.aufirst=J.-Y.&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.4064%252Faa-80-3-237-248&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABaum%E2%80%93Sweet+sequence" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Allouche and Shallit (2003) p 210.</span> </li> <li id="cite_note-FinAut-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-FinAut_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FinAut_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAllouche1993" class="citation journal cs1">Allouche, J.-.P. (1993). "Finite automata and arithmetic". <i>Séminaire Lotharingien de Combinatoire</i>: 23.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=S%C3%A9minaire+Lotharingien+de+Combinatoire&amp;rft.atitle=Finite+automata+and+arithmetic.&amp;rft.pages=23&amp;rft.date=1993&amp;rft.aulast=Allouche&amp;rft.aufirst=J.-.P.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABaum%E2%80%93Sweet+sequence" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Allouche and Shallit (2003) p 176.</span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Baum%E2%80%93Sweet_sequence&amp;action=edit&amp;section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlloucheShallit2003" class="citation book cs1">Allouche, Jean-Paul; <a href="/wiki/Jeffrey_Shallit" title="Jeffrey Shallit">Shallit, Jeffrey</a> (2003). <i>Automatic Sequences: Theory, Applications, Generalizations</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-521-82332-6" title="Special:BookSources/978-0-521-82332-6"><bdi>978-0-521-82332-6</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a>&#160;<a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&amp;q=an:1086.11015">1086.11015</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Automatic+Sequences%3A+Theory%2C+Applications%2C+Generalizations&amp;rft.pub=Cambridge+University+Press&amp;rft.date=2003&amp;rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1086.11015%23id-name%3DZbl&amp;rft.isbn=978-0-521-82332-6&amp;rft.aulast=Allouche&amp;rft.aufirst=Jean-Paul&amp;rft.au=Shallit%2C+Jeffrey&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABaum%E2%80%93Sweet+sequence" class="Z3988"></span></li></ul> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐6b7f745dd4‐c895w Cached time: 20241125131939 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.241 seconds Real time usage: 0.366 seconds Preprocessor visited node count: 797/1000000 Post‐expand include size: 15106/2097152 bytes Template argument size: 592/2097152 bytes Highest expansion depth: 8/100 Expensive parser function count: 1/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 30228/5000000 bytes Lua time usage: 0.130/10.000 seconds Lua memory usage: 4655854/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 254.997 1 -total 58.44% 149.033 1 Template:Reflist 36.69% 93.563 1 Template:MathWorld 30.52% 77.825 1 Template:Short_description 18.30% 46.665 2 Template:Pagetype 8.79% 22.405 4 Template:Cite_journal 7.36% 18.757 3 Template:Main_other 6.73% 17.170 3 Template:Cite_book 6.41% 16.355 1 Template:SDcat 3.95% 10.066 1 Template:Refbegin --> <!-- Saved in parser cache with key enwiki:pcache:20236431:|#|:idhash:canonical and timestamp 20241125131939 and revision id 1213358959. 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