Fibonacci Facts and Formulas
Identifieur interne : 000250 ( Istex/Corpus ); précédent : 000249; suivant : 000251Fibonacci Facts and Formulas
Auteurs : R. M. Capocelli ; G. Cerbone ; P. Cull ; J. L. HollowaySource :
Abstract
Abstract: We investigate several methods of computing Fibonacci numbers quickly and generalize some properties of the Fibonacci numbers to degree r Fibonacci (R-nacci) numbers. Sections 2 and 3 present several algorithms for computing the traditional, degree two, Fibonacci numbers quickly. Sections 4 and 5 investigate the structure of the binary representation of the Fibonacci numbers. Section 6 shows how the generalized Fibonacci numbers can be expressed as rounded powers of the dominant root of the characteristic equation. Properties of the roots of the characteristic equation of the generalized Fibonacci numbers are presented in Section 7. Section 8 introduces several properties of the Zeckendorf representation of the integers. Finally, in Section 9 the asymptotic proportion of l’s in the Zeckendorf representation of integers is computed and an easy to compute closed formula is given.
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DOI: 10.1007/978-1-4612-3352-7_9
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ISTEX:ED0710629A41A59B4F0D8239054970CDAB13774DLe document en format XML
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Capocelli</name><affiliation><mods:affiliation>Dipartimento di Informatica ed Applocazioni, Universita’ di Salerno, 84100, Salerno, Italy</mods:affiliation></affiliation></author><author><name sortKey="Cerbone, G" sort="Cerbone, G" uniqKey="Cerbone G" first="G." last="Cerbone">G. Cerbone</name><affiliation><mods:affiliation>Department of Computer Science, Oregon State University, 97331, Corvallis, Oregon, USA</mods:affiliation></affiliation></author><author><name sortKey="Cull, P" sort="Cull, P" uniqKey="Cull P" first="P." last="Cull">P. Cull</name><affiliation><mods:affiliation>Department of Computer Science, Oregon State University, 97331, Corvallis, Oregon, USA</mods:affiliation></affiliation></author><author><name sortKey="Holloway, J L" sort="Holloway, J L" uniqKey="Holloway J" first="J. L." last="Holloway">J. L. 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Section 6 shows how the generalized Fibonacci numbers can be expressed as rounded powers of the dominant root of the characteristic equation. Properties of the roots of the characteristic equation of the generalized Fibonacci numbers are presented in Section 7. Section 8 introduces several properties of the Zeckendorf representation of the integers. 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Sections 2 and 3 present several algorithms for computing the traditional, degree two, Fibonacci numbers quickly. Sections 4 and 5 investigate the structure of the binary representation of the Fibonacci numbers. Section 6 shows how the generalized Fibonacci numbers can be expressed as rounded powers of the dominant root of the characteristic equation. Properties of the roots of the characteristic equation of the generalized Fibonacci numbers are presented in Section 7. Section 8 introduces several properties of the Zeckendorf representation of the integers. 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TocLevels="0"><PartID>2</PartID><PartSequenceNumber>2</PartSequenceNumber><PartTitle>Combinatorics</PartTitle><PartChapterCount>16</PartChapterCount><PartContext><BookID>978-1-4612-3352-7</BookID><BookTitle>Sequences</BookTitle></PartContext></PartInfo><Chapter ID="Chap9" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" Language="En" NumberingStyle="Unnumbered" OutputMedium="All" TocLevels="0"><ChapterID>9</ChapterID><ChapterDOI>10.1007/978-1-4612-3352-7_9</ChapterDOI><ChapterSequenceNumber>9</ChapterSequenceNumber><ChapterTitle Language="En">Fibonacci Facts and Formulas</ChapterTitle><ChapterFirstPage>123</ChapterFirstPage><ChapterLastPage>137</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag New York Inc.</CopyrightHolderName><CopyrightYear>1990</CopyrightYear></ChapterCopyright><ChapterHistory><RegistrationDate><Year>2011</Year><Month>8</Month><Day>30</Day></RegistrationDate></ChapterHistory><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><PartID>2</PartID><BookID>978-1-4612-3352-7</BookID><BookTitle>Sequences</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff2"><AuthorName DisplayOrder="Western"><GivenName>R.</GivenName><GivenName>M.</GivenName><FamilyName>Capocelli</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff3"><AuthorName DisplayOrder="Western"><GivenName>G.</GivenName><FamilyName>Cerbone</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff3"><AuthorName DisplayOrder="Western"><GivenName>P.</GivenName><FamilyName>Cull</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff3"><AuthorName DisplayOrder="Western"><GivenName>J.</GivenName><GivenName>L.</GivenName><FamilyName>Holloway</FamilyName></AuthorName></Author><Affiliation ID="Aff2"><OrgDivision>Dipartimento di Informatica ed Applocazioni</OrgDivision><OrgName>Universita’ di Salerno</OrgName><OrgAddress><City>Salerno</City><Postcode>84100</Postcode><Country>Italy</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgDivision>Department of Computer Science</OrgDivision><OrgName>Oregon State University</OrgName><OrgAddress><City>Corvallis</City><State>Oregon</State><Postcode>97331</Postcode><Country>USA</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En" OutputMedium="Online"><Heading>Abstract</Heading><Para>We investigate several methods of computing Fibonacci numbers quickly and generalize some properties of the Fibonacci numbers to degree <Emphasis Type="Italic">r</Emphasis> Fibonacci (R-nacci) numbers. Sections 2 and 3 present several algorithms for computing the traditional, degree two, Fibonacci numbers quickly. Sections 4 and 5 investigate the structure of the binary representation of the Fibonacci numbers. Section 6 shows how the generalized Fibonacci numbers can be expressed as rounded powers of the dominant root of the characteristic equation. Properties of the roots of the characteristic equation of the generalized Fibonacci numbers are presented in Section 7. Section 8 introduces several properties of the Zeckendorf representation of the integers. Finally, in Section 9 the asymptotic proportion of l’s in the Zeckendorf representation of integers is computed and an easy to compute closed formula is given.</Para></Abstract><ArticleNote Type="Misc"><SimplePara>Supported by a fellowship from the Italian National Research Council (CNR 203.01.43)</SimplePara></ArticleNote></ChapterHeader><NoBody></NoBody></Chapter></Part></Book></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Fibonacci Facts and Formulas</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Fibonacci Facts and Formulas</title></titleInfo><name type="personal"><namePart type="given">R.</namePart><namePart type="given">M.</namePart><namePart type="family">Capocelli</namePart><affiliation>Dipartimento di Informatica ed Applocazioni, Universita’ di Salerno, 84100, Salerno, Italy</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">G.</namePart><namePart type="family">Cerbone</namePart><affiliation>Department of Computer Science, Oregon State University, 97331, Corvallis, Oregon, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">P.</namePart><namePart type="family">Cull</namePart><affiliation>Department of Computer Science, Oregon State University, 97331, Corvallis, Oregon, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">J.</namePart><namePart type="given">L.</namePart><namePart type="family">Holloway</namePart><affiliation>Department of Computer Science, Oregon State University, 97331, Corvallis, Oregon, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Renato</namePart><namePart type="given">M.</namePart><namePart type="family">Capocelli</namePart><affiliation>Dipartimento di Matematica, Università di Roma “La Sapienza”, I-00185, Rome, Italy</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="research-article" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer New York</publisher><place><placeTerm type="text">New York, NY</placeTerm></place><dateIssued encoding="w3cdtf">1990</dateIssued><copyrightDate encoding="w3cdtf">1990</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: We investigate several methods of computing Fibonacci numbers quickly and generalize some properties of the Fibonacci numbers to degree r Fibonacci (R-nacci) numbers. Sections 2 and 3 present several algorithms for computing the traditional, degree two, Fibonacci numbers quickly. Sections 4 and 5 investigate the structure of the binary representation of the Fibonacci numbers. Section 6 shows how the generalized Fibonacci numbers can be expressed as rounded powers of the dominant root of the characteristic equation. Properties of the roots of the characteristic equation of the generalized Fibonacci numbers are presented in Section 7. Section 8 introduces several properties of the Zeckendorf representation of the integers. Finally, in Section 9 the asymptotic proportion of l’s in the Zeckendorf representation of integers is computed and an easy to compute closed formula is given.</abstract><relatedItem type="host"><titleInfo><title>Sequences</title><subTitle>Combinatorics, Compression, Security, and Transmission</subTitle></titleInfo><name type="personal"><namePart type="given">Renato</namePart><namePart type="given">M.</namePart><namePart type="family">Capocelli</namePart><affiliation>Dipartimento di Matematica, Università di Roma “La Sapienza”, I-00185, Rome, Italy</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-5WTPMB5N-F">book</genre><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">1990</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11649">Mathematics and Statistics</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="SCM">Mathematics</topic><topic authority="SpringerSubjectCodes" authorityURI="M14068">Mathematical Modeling and Industrial Mathematics</topic><topic authority="SpringerSubjectCodes" authorityURI="M14050">Numerical Analysis</topic><topic authority="SpringerSubjectCodes" authorityURI="I16005">Theory of Computation</topic><topic authority="SpringerSubjectCodes" authorityURI="I15041">Coding and Information Theory</topic><topic authority="SpringerSubjectCodes" authorityURI="M29010">Combinatorics</topic><topic authority="SpringerSubjectCodes" authorityURI="T11006">Appl.Mathematics/Computational Methods of Engineering</topic></subject><identifier type="DOI">10.1007/978-1-4612-3352-7</identifier><identifier type="ISBN">978-1-4612-7977-8</identifier><identifier type="eISBN">978-1-4612-3352-7</identifier><identifier type="BookTitleID">23831</identifier><identifier type="BookID">978-1-4612-3352-7</identifier><identifier type="BookChapterCount">42</identifier><identifier type="PartChapterCount">16</identifier><part><date>1990</date><detail type="part"><title>Combinatorics</title></detail><extent unit="pages"><start>123</start><end>137</end></extent></part><recordInfo><recordOrigin>Springer-Verlag New York, 1990</recordOrigin></recordInfo></relatedItem><identifier type="istex">ED0710629A41A59B4F0D8239054970CDAB13774D</identifier><identifier type="ark">ark:/67375/HCB-65HLLFPK-K</identifier><identifier type="DOI">10.1007/978-1-4612-3352-7_9</identifier><identifier type="ChapterID">9</identifier><identifier type="ChapterID">Chap9</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag New York, 1990</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-RLRX46XW-4">springer</recordContentSource><recordOrigin>Springer-Verlag New York Inc., 1990</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/ark:/67375/HCB-65HLLFPK-K/record.json</uri></json:item></metadata><annexes><json:item><extension>txt</extension><original>true</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/ark:/67375/HCB-65HLLFPK-K/annexes.txt</uri></json:item></annexes><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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