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Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

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dc.contributor.authorBilgici Göksal
dc.contributor.authorŞentürk Tuncay Deniz
dc.date.accessioned2026-01-04T13:20:44Z
dc.date.issued2019-07-18
dc.description.abstractAbstract In this paper, we obtain a closed form for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>k</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${F_{\sum\nolimits_{i = 1}^k {} }}$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>k</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${P_{\sum\nolimits_{i = 1}^k {} }}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>k</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${J_{\sum\nolimits_{i = 1}^k {} }}$ for some positive integers k where Fr, Pr and Jr are the rth Fibonacci, Pell and Jacobsthal numbers, respectively. We also give three open problems for the general cases <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${F_{\sum\nolimits_{i = 1}^n {} }}$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${P_{\sum\nolimits_{i = 1}^n {} }}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${J_{\sum\nolimits_{i = 1}^n {} }}$ for any arbitrary positive integer n.
dc.description.urihttps://doi.org/10.2478/amsil-2019-0005
dc.description.urihttps://www.sciendo.com/pdf/10.2478/amsil-2019-0005
dc.description.urihttps://doaj.org/article/d525303f3fa0420594152db006e72144
dc.description.urihttps://dx.doi.org/10.2478/amsil-2019-0005
dc.identifier.doi10.2478/amsil-2019-0005
dc.identifier.eissn2391-4238
dc.identifier.endpage65
dc.identifier.issn0860-2107
dc.identifier.openairedoi_dedup___::daa8254aa243618d75167a676215eb39
dc.identifier.orcid0000-0001-9964-5578
dc.identifier.orcid0000-0002-5247-5097
dc.identifier.scopus2-s2.0-85095690560
dc.identifier.startpage55
dc.identifier.urihttps://hdl.handle.net/20.500.12597/37430
dc.identifier.volume33
dc.identifier.wos000644474000004
dc.language.isoeng
dc.publisherWalter de Gruyter GmbH
dc.relation.ispartofAnnales Mathematicae Silesianae
dc.rightsOPEN
dc.subjectJacobsthal numbers
dc.subjectQA1-939
dc.subjectB39
dc.subjectK31
dc.subjectY55
dc.subjectFibonacci numbers
dc.subjectPell numbers
dc.subjectMathematics
dc.titleSome Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers
dc.typeArticle
dspace.entity.typePublication
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We also give three open problems for the general cases <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_amsil-2019-0005_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:tex-math>${F_{\\sum\\nolimits_{i = 1}^n {} }}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_amsil-2019-0005_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:tex-math>${P_{\\sum\\nolimits_{i = 1}^n {} }}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_amsil-2019-0005_eq_006.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:tex-math>${J_{\\sum\\nolimits_{i = 1}^n {} }}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>for any arbitrary positive integer <jats:italic>n</jats:italic>.</jats:p>"],"publicationDate":"2019-07-18","publisher":"Walter de Gruyter GmbH","embargoEndDate":null,"sources":["Crossref","Annales Mathematicae Silesianae, Vol 33, Iss 1, Pp 55-65 (2019)"],"formats":null,"contributors":null,"coverages":null,"bestAccessRight":{"code":"c_abf2","label":"OPEN","scheme":"http://vocabularies.coar-repositories.org/documentation/access_rights/"},"container":{"name":"Annales Mathematicae Silesianae","issnPrinted":"0860-2107","issnOnline":"2391-4238","issnLinking":null,"ep":"65","iss":null,"sp":"55","vol":"33","edition":null,"conferencePlace":null,"conferenceDate":null},"documentationUrls":null,"codeRepositoryUrl":null,"programmingLanguage":null,"contactPeople":null,"contactGroups":null,"tools":null,"size":null,"version":null,"geoLocations":null,"id":"doi_dedup___::daa8254aa243618d75167a676215eb39","originalIds":["10.2478/amsil-2019-0005","50|doiboost____|daa8254aa243618d75167a676215eb39","50|doajarticles::9ac8241faa215f03bb9c66290cb52b6a","oai:doaj.org/article:d525303f3fa0420594152db006e72144","2950685760"],"pids":[{"scheme":"doi","value":"10.2478/amsil-2019-0005"}],"dateOfCollection":null,"lastUpdateTimeStamp":null,"indicators":{"citationImpact":{"citationCount":2,"influence":3.2116576e-9,"popularity":2.2616646e-9,"impulse":2,"citationClass":"C5","influenceClass":"C5","impulseClass":"C5","popularityClass":"C5"}},"instances":[{"pids":[{"scheme":"doi","value":"10.2478/amsil-2019-0005"}],"license":"CC BY NC ND","type":"Article","urls":["https://doi.org/10.2478/amsil-2019-0005"],"publicationDate":"2019-07-18","refereed":"peerReviewed"},{"pids":[{"scheme":"doi","value":"10.2478/amsil-2019-0005"}],"license":"CC BY","type":"Article","urls":["https://www.sciendo.com/pdf/10.2478/amsil-2019-0005"],"refereed":"nonPeerReviewed"},{"alternateIdentifiers":[{"scheme":"doi","value":"10.2478/amsil-2019-0005"}],"type":"Article","urls":["https://doaj.org/article/d525303f3fa0420594152db006e72144"],"publicationDate":"2019-09-01","refereed":"nonPeerReviewed"},{"alternateIdentifiers":[{"scheme":"mag_id","value":"2950685760"},{"scheme":"doi","value":"10.2478/amsil-2019-0005"}],"type":"Other literature type","urls":["https://dx.doi.org/10.2478/amsil-2019-0005"],"refereed":"nonPeerReviewed"}],"isGreen":false,"isInDiamondJournal":true}
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