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<title>Negafibonacci coding</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Negafibonacci coding</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <b>negafibonacci coding</b> is a <a href="Universal_code_(data_compression)" title="Universal code (data compression)">universal code</a> which encodes nonzero integers into binary <a href="Code_word_(communication)" title="Code word (communication)">code words</a>. It is similar to <a href="Fibonacci_coding" title="Fibonacci coding">Fibonacci coding</a>, except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end.

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<div class="mw-heading mw-heading2"><h2 id="Encoding_method">Encoding method</h2></div>

<p>The following steps describe how to encode a nonzero integer <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 x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>. Note that <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 f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> denotes the Negafibonacci sequence.

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> is positive, compute the greatest odd negative integer <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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</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> such that the sum of the odd negative terms of the Negafibonacci sequence from −1 to <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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</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> with a step of −2, is greater than or equal to <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 x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>: <br> <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\in \{-\left(2k+1\right),k\in [0,\infty [\},\quad \sum _{i=-1,\;i\;odd}^{n-2}f(i)<x\leq \sum _{i=-1,\;i\;odd}^{n}f(i).}">

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<annotation encoding="application/x-tex">{\displaystyle n\in \{-\left(2k+1\right),k\in [0,\infty [\},\quad \sum _{i=-1,\;i\;odd}^{n-2}f(i)<x\leq \sum _{i=-1,\;i\;odd}^{n}f(i).}</annotation>

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</math></span><img src="./389d9659311f58a87e73951de1d5e17fd319a328.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:65.87ex; height:7.676ex;" alt="{\displaystyle n\in \{-\left(2k+1\right),k\in [0,\infty [\},\quad \sum _{i=-1,\;i\;odd}^{n-2}f(i)<x\leq \sum _{i=-1,\;i\;odd}^{n}f(i).}" loading="lazy"></span> <br> 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 x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> is negative, compute the greatest even negative integer <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}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</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> such that the sum of the even negative terms of the Negafibonacci sequence from 0 to <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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</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> with a step of −2, is less than or equal to <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 x}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>: <br> <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\in \{-2k,k\in [2,\infty [\},\quad \sum _{i=-2,\;i\;even}^{n-2}f(i)>x\geq \sum _{i=-2,\;i\;even}^{n}f(i)}">

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<annotation encoding="application/x-tex">{\displaystyle n\in \{-2k,k\in [2,\infty [\},\quad \sum _{i=-2,\;i\;even}^{n-2}f(i)>x\geq \sum _{i=-2,\;i\;even}^{n}f(i)}</annotation>

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</math></span><img src="./5bdb38d1fe0f759300a5be8cf43137cb1b7b4509.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:60.235ex; height:7.676ex;" alt="{\displaystyle n\in \{-2k,k\in [2,\infty [\},\quad \sum _{i=-2,\;i\;even}^{n-2}f(i)>x\geq \sum _{i=-2,\;i\;even}^{n}f(i)}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle |n|^{\text{th}}}</annotation>

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</math></span><img src="./1cb66892831beba144f2e13d0b047de401771ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.474ex; height:3.343ex;" alt="{\displaystyle |n|^{\text{th}}}" loading="lazy"></span> bit of the binary word. Subtract <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 f(n)}">

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<annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation>

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</math></span><img src="./c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" loading="lazy"></span> from <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 x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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<li>Repeat the process from step 1 with the new value of <i>x</i>, until it reaches 0.</li>

<li>Add a 1 on the left of the resulting binary word to finish the encoding.</li></ol>

<p>To decode an encoded binary word, remove the leftmost 1 from the binary word, since it is used only to denote the end of the encoded number. Then assign the remaining bits the values of the Negafibonacci sequence from −1 (1, −1, 2, −3, 5, −8, 13...), and sum the all the values associated with a 1.

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<div class="mw-heading mw-heading2"><h2 id="Negafibonacci_representation">Negafibonacci representation</h2></div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Numeral_system" title="Numeral system">Numeral systems</a></th></tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Positional_notation" title="Positional notation">Place-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numerals</a></div></div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="Arabic_numerals" title="Arabic numerals">Western Arabic</a></li>

<li><a href="Eastern_Arabic_numerals" title="Eastern Arabic numerals">Eastern Arabic</a></li></ul>

<hr>

<ul><li><a href="Bengali_numerals" title="Bengali numerals">Bengali</a></li>

<li><a href="Devanagari_numerals" title="Devanagari numerals">Devanagari</a></li>

<li><a href="Gujarati_numerals" title="Gujarati numerals">Gujarati</a></li>

<li><a href="Gurmukhi_numerals" class="mw-redirect" title="Gurmukhi numerals">Gurmukhi</a></li>

<li><a href="Sinhala_numerals" title="Sinhala numerals">Sinhala</a></li>

<li><a href="Tamil_numerals" title="Tamil numerals">Tamil</a></li>

<li><a href="Malayalam_numerals" title="Malayalam numerals">Malayalam</a></li>

<li><a href="Telugu_script#Numerals" title="Telugu script">Telugu</a></li>

<li><a href="Kannada_script#Numerals" title="Kannada script">Kannada</a></li>

<li><a href="Dzongkha_numerals" title="Dzongkha numerals">Dzongkha</a></li></ul>

<hr>

<ul><li><a href="Tibetan_numerals" title="Tibetan numerals">Tibetan</a></li>

<li><a href="Balinese_numerals" title="Balinese numerals">Balinese</a></li>

<li><a href="Burmese_numerals" title="Burmese numerals">Burmese</a></li>

<li><a href="Javanese_numerals" title="Javanese numerals">Javanese</a></li>

<li><a href="Khmer_numerals" title="Khmer numerals">Khmer</a></li>

<li><a href="Mongolian_numerals" title="Mongolian numerals">Mongolian</a></li>

<li><a href="Sundanese_numerals" title="Sundanese numerals">Sundanese</a></li>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">East Asian systems</div></div><div class="sidebar-list-content mw-collapsible-content">

<dl><dt>Contemporary</dt></dl>

<ul><li><a href="Chinese_numerals" title="Chinese numerals">Chinese</a>

<ul><li><a href="Hokkien_numerals" title="Hokkien numerals">Hokkien</a></li>

<li><a href="Suzhou_numerals" title="Suzhou numerals">Suzhou</a></li></ul></li>

<li><a href="Japanese_numerals" title="Japanese numerals">Japanese</a></li>

<li><a href="Korean_numerals" title="Korean numerals">Korean</a></li>

<li><a href="Vietnamese_numerals" title="Vietnamese numerals">Vietnamese</a></li></ul>

<hr>

<dl><dt>Historic</dt></dl>

<ul><li><a href="Counting_rods" title="Counting rods">Counting rods</a></li>

<li><a href="Tangut_numerals" title="Tangut numerals">Tangut</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Other systems</div></div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="History_of_ancient_numeral_systems" title="History of ancient numeral systems">History</a></li></ul>

<hr>

<dl><dt><a href="Ancient_history" title="Ancient history">Ancient</a></dt></dl>

<ul><li><a href="Babylonian_cuneiform_numerals" title="Babylonian cuneiform numerals">Babylonian</a></li></ul>

<hr>

<dl><dt><a href="Post-classical_history" title="Post-classical history">Post-classical</a></dt></dl>

<ul><li><a href="Cistercian_numerals" title="Cistercian numerals">Cistercian</a></li>

<li><a href="Maya_numerals" title="Maya numerals">Mayan</a></li>

<li><a href="Muisca_numerals" title="Muisca numerals">Muisca</a></li>

<li><a href="Pentadic_numerals" title="Pentadic numerals">Pentadic</a></li>

<li><a href="Quipu" title="Quipu">Quipu</a></li>

<hr>

<dl><dt>Contemporary</dt></dl>

<ul><li><a href="Cherokee_syllabary#Numerals" title="Cherokee syllabary">Cherokee</a></li>

<li><a href="Kaktovik_numerals" title="Kaktovik numerals">Kaktovik</a> (Iñupiaq)</li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">By <a href="Radix" title="Radix">radix/base</a></div></div><div class="sidebar-list-content mw-collapsible-content">

<dl><dt>Common radices/bases</dt></dl>

<hr>

<dl><dt><a href="Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard radices/bases</a></dt></dl>

<ul><li><a href="Bijective_numeration" title="Bijective numeration">Bijective</a><span class="nowrap"> </span>(<a href="Unary_numeral_system" title="Unary numeral system">1</a>)</li>

<li><a href="Signed-digit_representation" title="Signed-digit representation">Signed-digit</a><span class="nowrap"> </span>(<a href="Balanced_ternary" title="Balanced ternary">balanced ternary</a>)</li>

<li><a href="Mixed_radix" title="Mixed radix">Mixed</a><span class="nowrap"> </span>(<a href="Factorial_number_system" title="Factorial number system">factorial</a>)</li>

<li><a href="Negative_base" title="Negative base">Negative</a></li>

<li><a href="Complex-base_system" title="Complex-base system">Complex</a><span class="nowrap"> </span>(<a href="Quater-imaginary_base" title="Quater-imaginary base">2<i>i</i></a>)</li>

<li><a href="Non-integer_base_of_numeration" title="Non-integer base of numeration">Non-integer</a><span class="nowrap"> </span>(<a href="Golden_ratio_base" title="Golden ratio base">φ</a>)</li>

<li><a href="Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric</a></li></ul></div></div></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Sign-value_notation" title="Sign-value notation">Sign-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content">

<dl><dt>Non-alphabetic</dt></dl>

<ul><li><a href="Aegean_numerals" title="Aegean numerals">Aegean</a></li>

<li><a href="Attic_numerals" title="Attic numerals">Attic</a></li>

<li><a href="Aztec_script#Numerals" title="Aztec script">Aztec</a></li>

<li><a href="Brahmi_numerals" title="Brahmi numerals">Brahmi</a></li>

<li><a href="Chuvash_numerals" title="Chuvash numerals">Chuvash</a></li>

<li><a href="Egyptian_numerals" title="Egyptian numerals">Egyptian</a></li>

<li><a href="Etruscan_numerals" title="Etruscan numerals">Etruscan</a></li>

<li><a href="Kharosthi_numerals" class="mw-redirect" title="Kharosthi numerals">Kharosthi</a></li>

<li><a href="Prehistoric_counting" class="mw-redirect" title="Prehistoric counting">Prehistoric counting</a></li>

<li><a href="Proto-cuneiform" title="Proto-cuneiform">Proto-cuneiform</a></li>

<li><a href="Roman_numerals" title="Roman numerals">Roman</a></li>

<li><a href="Tally_marks" title="Tally marks">Tally marks</a></li></ul>

<hr>

<dl><dt><a href="Alphabetic_numeral_system" title="Alphabetic numeral system">Alphabetic</a></dt></dl>

<ul><li><a href="Abjad_numerals" title="Abjad numerals">Abjad</a></li>

<li><a href="Armenian_numerals" title="Armenian numerals">Armenian</a></li>

<li><a href="Alphasyllabic_numeral_system" title="Alphasyllabic numeral system">Alphasyllabic</a>

<ul><li><a href="Aksharapalli" title="Aksharapalli">Akṣarapallī</a></li>

<li><a href="%C4%80ryabha%E1%B9%ADa_numeration" title="Āryabhaṭa numeration">Āryabhaṭa</a></li>

<li><a href="Katapayadi_system" title="Katapayadi system">Kaṭapayādi</a></li></ul></li>

<li><a href="Coptic_numerals" class="mw-redirect" title="Coptic numerals">Coptic</a></li>

<li><a href="Cyrillic_numerals" title="Cyrillic numerals">Cyrillic</a></li>

<li><a href="Ge%CA%BDez_script#Numerals" title="Geʽez script">Geʽez</a></li>

<li><a href="Georgian_numerals" title="Georgian numerals">Georgian</a></li>

<li><a href="Glagolitic_numerals" title="Glagolitic numerals">Glagolitic</a></li>

<li><a href="Greek_numerals" title="Greek numerals">Greek</a></li>

<li><a href="Hebrew_numerals" title="Hebrew numerals">Hebrew</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-below" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;">

<a href="List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></td></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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<p>Negafibonacci coding is closely related to <b>negafibonacci representation</b>, a positional <a href="Numeral_system" title="Numeral system">numeral system</a> sometimes used by mathematicians. The negafibonacci code for a particular nonzero integer is exactly that of the integer's negafibonacci representation, except with the order of its digits reversed and an additional "1" appended to the end. The negafibonacci code for all negative numbers has an odd number of digits, while those of all positive numbers have an even number of digits.

</p>

<div class="mw-heading mw-heading2"><h2 id="Table">Table</h2></div>

<p>The code for the integers from −11 to 11 is given below.

</p>

<th>Number

</th>

<th>Negafibonacci representation

</th>

<th>Negafibonacci code

</th></tr>

<tr>

<td>0001011

</td></tr>

<tr>

<td>1001011

</td></tr>

<tr>

<td>0100011

</td></tr>

<tr>

<td>0000011

</td></tr>

<tr>

<td>1000011

</td></tr>

<tr>

<td>0010011

</td></tr>

<tr>

<td>1010011

</td></tr>

<tr>

<td>01011

</td></tr>

<tr>

<td>00011

</td></tr>

<tr>

<td>10011

</td></tr>

<tr>

<td>011

</td></tr>

<tr>

<td>(cannot be encoded)

</td></tr>

<tr>

<td>11

</td></tr>

<tr>

<td>0011

</td></tr>

<tr>

<td>1011

</td></tr>

<tr>

<td>010011

</td></tr>

<tr>

<td>000011

</td></tr>

<tr>

<td>100011

</td></tr>

<tr>

<td>001011

</td></tr>

<tr>

<td>101011

</td></tr>

<tr>

<td>01010011

</td></tr>

<tr>

<td>00010011

</td></tr>

<tr>

<td>10010011

</td></tr></tbody></table>

<ul><li><a href="Fibonacci_numbers" class="mw-redirect" title="Fibonacci numbers">Fibonacci numbers</a></li>

<li><a href="Golden_ratio_base" title="Golden ratio base">Golden ratio base</a></li>

<li><a href="Zeckendorf's_theorem" title="Zeckendorf's theorem">Zeckendorf's theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</div>

<div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3></div>

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</style><cite id="CITEREFKnuth2008" class="citation conference cs1">Knuth, Donald (2008). <i>Negafibonacci Numbers and the Hyperbolic Plane</i>. Annual meeting of the Mathematical Association of America. San Jose, California.</cite></li>

<li><cite id="CITEREFKnuth2009" class="citation book cs1">Knuth, Donald (2009). <i><a href="The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>, Volume 4, Fascicle 1: Bitwise Tricks & Techniques; Binary Decision Diagrams</i>. Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-321-58050-4</bdi>.</cite> In the <a rel="nofollow" class="external text" href="https://www-cs-faculty.stanford.edu/~uno/fasc1a.ps.gz">pre-publication draft of section 7.1.3</a> see in particular pp. 36–39.</li>

<li><cite id="CITEREFMargenstern2008" class="citation book cs1">Margenstern, Maurice (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eEgvfic3A4kC&pg=PA79"><i>Cellular Automata in Hyperbolic Spaces</i></a>. Advances in unconventional computing and cellular automata. Vol. 2. Archives contemporaines. p. 79. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9782914610834</bdi>.</cite></li></ul>

</div>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Data_compression_methods241" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_compression_methods241" style="font-size:114%;margin:0 4em"><a href="Data_compression" title="Data compression">Data compression</a> methods</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lossless_compression" title="Lossless compression">Lossless</a><br>type</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Entropy_coding" title="Entropy coding">Entropy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_coding" title="Adaptive coding">Adaptive coding</a></li>

<li><a href="Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li>

<li><a href="Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric numeral systems</a></li>

<li><a href="Golomb_coding" title="Golomb coding">Golomb</a></li>

<li><a href="Huffman_coding" title="Huffman coding">Huffman</a>

<ul><li><a href="Adaptive_Huffman_coding" title="Adaptive Huffman coding">Adaptive</a></li>

<li><a href="Canonical_Huffman_code" title="Canonical Huffman code">Canonical</a></li>

<li><a href="Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li>

<li><a href="Range_coding" title="Range coding">Range</a></li>

<li><a href="Shannon_coding" title="Shannon coding">Shannon</a></li>

<li><a href="Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a></li>

<li><a href="Shannon%E2%80%93Fano%E2%80%93Elias_coding" title="Shannon–Fano–Elias coding">Shannon–Fano–Elias</a></li>

<li><a href="Tunstall_coding" title="Tunstall coding">Tunstall</a></li>

<li><a href="Unary_coding" title="Unary coding">Unary</a></li>

<li><a href="Universal_code_(data_compression)" title="Universal code (data compression)">Universal</a>

<ul><li><a href="Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a></li>

<li><a href="Fibonacci_coding" title="Fibonacci coding">Fibonacci</a></li>

<li><a href="Elias_gamma_coding" title="Elias gamma coding">Gamma</a></li>

<li><a href="Levenshtein_coding" title="Levenshtein coding">Levenshtein</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Dictionary_coder" title="Dictionary coder">Dictionary</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Byte-pair_encoding" title="Byte-pair encoding">Byte-pair encoding</a></li>

<li><a href="LZ77_and_LZ78" title="LZ77 and LZ78">Lempel–Ziv</a>

<li><a href="Snappy_(compression)" title="Snappy (compression)">Snappy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Delta_encoding" title="Delta encoding">Delta</a>

<ul><li><a href="Incremental_encoding" title="Incremental encoding">Incremental</a></li></ul></li>

<li><a href="Grammar-based_code" title="Grammar-based code">Grammar</a>

<li><a href="Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Hybrid</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>LZ77 + Huffman

<ul><li><a href="Deflate" title="Deflate">Deflate</a></li>

<li>LZ77 + ANS

<ul><li><a href="LZFSE" title="LZFSE">LZFSE</a></li></ul></li>

<li>LZ77 + Huffman + ANS

<ul><li><a href="Zstd" title="Zstd">Zstandard</a></li></ul></li>

<li>LZ77 + Huffman + context

<ul><li><a href="Brotli" title="Brotli">Brotli</a></li></ul></li>

<li>LZSS + Huffman

<li>LZ77 + Range

<li>LZHAM</li></ul></li>

<li>RLE + BWT + MTF + Huffman

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lossy_compression" title="Lossy compression">Lossy</a><br>type</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Transform_coding" title="Transform coding">Transform</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

<li><a href="Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Predictive</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li></ul></li>

<ul><li><a href="Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li>

<li>Motion

<ul><li><a href="Motion_compensation" title="Motion compensation">Compensation</a></li>

<li><a href="Motion_estimation" title="Motion estimation">Estimation</a></li>

<li><a href="Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li>

<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data_compression#Audio" title="Data compression">Audio</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Companding" title="Companding">Companding</a></li>

<li><a href="Convolution" title="Convolution">Convolution</a></li>

<li><a href="Dynamic_range" title="Dynamic range">Dynamic range</a></li>

<li><a href="Latency_(audio)" title="Latency (audio)">Latency</a></li>

<li><a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon theorem</a></li>

<li><a href="Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></li>

<li><a href="Silence_compression" title="Silence compression">Silence compression</a></li>

<li><a href="Sound_quality" title="Sound quality">Sound quality</a></li>

<li><a href="Speech_coding" title="Speech coding">Speech coding</a></li>

<li><a href="Sub-band_coding" title="Sub-band coding">Sub-band coding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Audio_codec" title="Audio codec">Codec</a><br>parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li>

<ul><li><a href="Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li>

<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Image_compression" title="Image compression">Image</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a></li>

<li><a href="Coding_tree_unit" title="Coding tree unit">Coding tree unit</a></li>

<li><a href="Color_space" title="Color space">Color space</a></li>

<li><a href="Compression_artifact" title="Compression artifact">Compression artifact</a></li>

<li><a href="Image_resolution" title="Image resolution">Image resolution</a></li>

<li><a href="Macroblock" title="Macroblock">Macroblock</a></li>

<li><a href="Pixel" title="Pixel">Pixel</a></li>

<li><a href="Quantization_(image_processing)" title="Quantization (image processing)">Quantization</a></li>

<li><a href="Standard_test_image" title="Standard test image">Standard test image</a></li>

<li><a href="Texture_compression" title="Texture compression">Texture compression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chain_code" title="Chain code">Chain code</a></li>

<li><a href="Deflate" title="Deflate">Deflate</a></li>

<li><a href="Fractal_compression" title="Fractal compression">Fractal</a></li>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

<li><a href="Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data_compression#Video" title="Data compression">Video</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Display_resolution" title="Display resolution">Display resolution</a></li>

<li><a href="Film_frame" title="Film frame">Frame</a></li>

<li><a href="Frame_rate" title="Frame rate">Frame rate</a></li>

<li><a href="Video_compression_picture_types" title="Video compression picture types">Frame types</a></li>

<li><a href="Interlaced_video" title="Interlaced video">Interlace</a></li>

<li><a href="Video#Characteristics_of_video_streams" title="Video">Video characteristics</a></li>

<li><a href="Video_quality" title="Video quality">Video quality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Video_codec" title="Video codec">Codec</a><br>parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Deblocking_filter" title="Deblocking filter">Deblocking filter</a></li>

<li><a href="Lapped_transform" title="Lapped transform">Lapped transform</a></li>

<li>Motion

<ul><li><a href="Motion_compensation" title="Motion compensation">Compensation</a></li>

<li><a href="Motion_estimation" title="Motion estimation">Estimation</a></li>

<li><a href="Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Information_theory" title="Information theory">Theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Compressed_data_structure" title="Compressed data structure">Compressed data structures</a>

<ul><li><a href="Compressed_suffix_array" title="Compressed suffix array">Compressed suffix array</a></li>

<li><a href="FM-index" title="FM-index">FM-index</a></li></ul></li>

<li><a href="Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a>

<ul><li><a href="Timeline_of_information_theory" title="Timeline of information theory">Timeline</a></li></ul></li>

<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>

<li><a href="Prefix_code" title="Prefix code">Prefix code</a></li>

<li><a href="Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li>

<li><a href="Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion</a></li>

<li><a href="Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li>

<li><a href="Data_compression_symmetry" title="Data compression symmetry">Symmetry</a></li>

<li><a href="Smallest_grammar_problem" title="Smallest grammar problem">Smallest grammar problem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Community</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hutter_Prize" title="Hutter Prize">Hutter Prize</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mark_Adler" title="Mark Adler">Mark Adler</a></li>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-12" href="https://en.wikipedia.org/wiki/?title=Negafibonacci_coding&oldid=1300161375">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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