{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T11:34:21Z","timestamp":1771673661733,"version":"3.50.1"},"reference-count":37,"publisher":"Elsevier BV","issue":"2","license":[{"start":{"date-parts":[[2022,2,1]],"date-time":"2022-02-01T00:00:00Z","timestamp":1643673600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2022,2,1]],"date-time":"2022-02-01T00:00:00Z","timestamp":1643673600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T00:00:00Z","timestamp":1643241600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cell.com","elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Heliyon"],"published-print":{"date-parts":[[2022,2]]},"DOI":"10.1016\/j.heliyon.2022.e08863","type":"journal-article","created":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T18:45:50Z","timestamp":1643827550000},"page":"e08863","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":11,"title":["The bi-periodic Horadam sequence and some perturbed tridiagonal 2-Toeplitz matrices: A unified approach"],"prefix":"10.1016","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3348-1141","authenticated-orcid":false,"given":"Milica","family":"An\u0111eli\u0107","sequence":"first","affiliation":[]},{"given":"Carlos M.","family":"da Fonseca","sequence":"additional","affiliation":[]},{"given":"Fatih","family":"Y\u0131lmaz","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.heliyon.2022.e08863_br0010","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1080\/00029890.2005.11920232","article-title":"Fibonacci, Chebyshev, and orthogonal polynomials","volume":"112","author":"Aharonov","year":"2005","journal-title":"Am. Math. Mon."},{"key":"10.1016\/j.heliyon.2022.e08863_br0020","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1080\/10236198.2019.1709180","article-title":"Second-order difference equations with sign-alternating coefficients","volume":"26","author":"An\u0111eli\u0107","year":"2020","journal-title":"J. Differ. Equ. Appl."},{"issue":"4","key":"10.1016\/j.heliyon.2022.e08863_br0030","doi-asserted-by":"crossref","first-page":"116","DOI":"10.7546\/nntdm.2021.27.4.116-121","article-title":"Determinantal representations for the number of subsequences without isolated odd terms","volume":"27","author":"An\u0111eli\u0107","year":"2021","journal-title":"Notes Number Theory Discrete Math."},{"issue":"8","key":"10.1016\/j.heliyon.2022.e08863_br0040","doi-asserted-by":"crossref","DOI":"10.1016\/j.heliyon.2021.e07764","article-title":"On the constant coefficients of a certain recurrence relation: a simple proof","volume":"7","author":"An\u0111eli\u0107","year":"2021","journal-title":"Heliyon"},{"key":"10.1016\/j.heliyon.2022.e08863_br0050","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.laa.2019.03.001","article-title":"Some comments on tridiagonal (p,r)-Toeplitz matrices","volume":"572","author":"An\u0111eli\u0107","year":"2019","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.heliyon.2022.e08863_br0060","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/j.laa.2010.09.017","article-title":"On the number of P-vertices of some graphs","volume":"434","author":"An\u0111eli\u0107","year":"2011","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.heliyon.2022.e08863_br0070","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00150517.1963.12431539","article-title":"Fibonacci numbers, Chebyshev polynomials generalizations and difference equations","volume":"1","author":"Buschman","year":"1963","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0080","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1016\/j.laa.2008.09.034","article-title":"Exact eigensystems for some matrices arising from discretizations","volume":"430","author":"Chang","year":"2009","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.heliyon.2022.e08863_br0090","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.3390\/math8091527","article-title":"Relations between generalized bi-periodic Fibonacci and Lucas sequences","volume":"8","author":"Choo","year":"2020","journal-title":"Mathematics"},{"key":"10.1016\/j.heliyon.2022.e08863_br0100","doi-asserted-by":"crossref","DOI":"10.26493\/1855-3974.2473.f2e","article-title":"New strong divisibility sequences","author":"Du","year":"2022","journal-title":"Ars Math. Contemp."},{"key":"10.1016\/j.heliyon.2022.e08863_br0110","series-title":"The characteristic roots of certain real symmetric matrices","author":"Elliott","year":"1953"},{"key":"10.1016\/j.heliyon.2022.e08863_br0120","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/BF02174148","article-title":"Remarks on band matrices","volume":"10","author":"Elsner","year":"1967","journal-title":"Numer. Math."},{"key":"10.1016\/j.heliyon.2022.e08863_br0130","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1080\/00150517.1978.12430296","article-title":"The Fibonacci pseudogroup, characteristic polynomials and eigenvalues of tridiagonal matrices, periodic linear recurrence systems and application to quantum mechanics","volume":"16","author":"Ferguson","year":"1978","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0140","first-page":"41","article-title":"Unifying some Pell and Fibonacci identities","volume":"236","author":"da Fonseca","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.heliyon.2022.e08863_br0150","first-page":"205","article-title":"A multi-temperature kinetic Ising model and the eigenvalues of some perturbed Jacobi matrices","volume":"259","author":"da Fonseca","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.heliyon.2022.e08863_br0160","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1007\/s10474-019-00970-1","article-title":"Eigenpairs of a family of tridiagonal matrices: three decades later","volume":"160","author":"da Fonseca","year":"2020","journal-title":"Acta Math. Hung."},{"issue":"3","key":"10.1016\/j.heliyon.2022.e08863_br0170","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1007\/s00211-005-0596-3","article-title":"Explicit inverse of a tridiagonal k-Toeplitz matrix","volume":"100","author":"da Fonseca","year":"2005","journal-title":"Numer. Math."},{"key":"10.1016\/j.heliyon.2022.e08863_br0180","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/S0024-3795(00)00289-5","article-title":"Explicit inverses of some tridiagonal matrices","volume":"325","author":"da Fonseca","year":"2001","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.heliyon.2022.e08863_br0190","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1093\/imanum\/5.1.101","article-title":"Inversion of Toeplitz matrices which are not strongly singular","volume":"5","author":"Gover","year":"1985","journal-title":"IMA J. Numer. Anal."},{"key":"10.1016\/j.heliyon.2022.e08863_br0200","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1080\/00150517.1996.12429084","article-title":"Numbers of subsequences without isolated odd members","volume":"34","author":"Guy","year":"1996","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0210","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1080\/00150517.1996.12429096","article-title":"Jacobsthal representations numbers","volume":"34","author":"Horadam","year":"1996","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0220","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1080\/00150517.1965.12431416","article-title":"Basic properties of a certain generalized sequence of numbers","volume":"3","author":"Horadam","year":"1965","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0230","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1080\/00150517.1965.12431416","article-title":"Special properties of the sequence Wn(a,b;p,q)","volume":"3","author":"Horadam","year":"1965","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0240","first-page":"437","article-title":"Generating functions for powers of a certain generalised sequence of numbers","volume":"68","author":"Horadam","year":"1965","journal-title":"Duke Math. J."},{"key":"10.1016\/j.heliyon.2022.e08863_br0250","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1080\/00150517.1975.12430668","article-title":"Fibonacci and related sequences in periodic tridiagonal matrices","volume":"13","author":"Lehmer","year":"1975","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0260","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/BF00051649","article-title":"Eigenvalues and eigenvectors of some tridiagonal matrices","volume":"60","author":"Losonczi","year":"1992","journal-title":"Acta Math. Hung."},{"key":"10.1016\/j.heliyon.2022.e08863_br0270","doi-asserted-by":"crossref","first-page":"1663","DOI":"10.1049\/piee.1963.0236","article-title":"Matrix analysis of transient voltage distributions in alternating ladder networks","volume":"110","author":"Lovass-Nagy","year":"1963","journal-title":"Proc. Inst. Electr. Eng."},{"key":"10.1016\/j.heliyon.2022.e08863_br0280","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/0024-3795(69)90030-5","article-title":"On periodic continuants","volume":"2","author":"R\u00f3zsa","year":"1969","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.heliyon.2022.e08863_br0290","first-page":"229","article-title":"Some continuant determinants arising in physics and chemistry","volume":"62","author":"Rutherford","year":"1947","journal-title":"Proc. R. Soc. Edinb., Sect. A"},{"key":"10.1016\/j.heliyon.2022.e08863_br0300","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1016\/S0747-7171(08)80160-5","article-title":"On the worst case of three algorithms for computing the Jacobi symbol","volume":"10","author":"Shallit","year":"1990","journal-title":"J. Symb. Comput."},{"key":"10.1016\/j.heliyon.2022.e08863_br0310","first-page":"1","article-title":"Alwyn Horadam: the man and his mathematics","volume":"22","author":"Shannon","year":"2016","journal-title":"Notes Number Theory Discrete Math."},{"key":"10.1016\/j.heliyon.2022.e08863_br0320","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1080\/00150517.1987.12429704","article-title":"Ad multos annos","volume":"25","author":"Shannon","year":"1987","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0330","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1080\/00150517.2020.12427607","article-title":"On some linear recurrences","volume":"73","author":"da Silva","year":"2020","journal-title":"Fibonacci Q."},{"key":"10.1016\/j.heliyon.2022.e08863_br0340","series-title":"The On-Line Encyclopedia of Integer Sequences","author":"Sloane","year":"2019"},{"key":"10.1016\/j.heliyon.2022.e08863_br0350","doi-asserted-by":"crossref","DOI":"10.1186\/s13662-020-2507-4","article-title":"Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences","volume":"2020","author":"Tan","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"10.1016\/j.heliyon.2022.e08863_br0360","first-page":"143","article-title":"A note on the sequence (Wn)n\u2a7e0 of A.F. Horadam","volume":"53","author":"Udrea","year":"1996","journal-title":"Port. Math."},{"key":"10.1016\/j.heliyon.2022.e08863_br0370","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/BF03024279","article-title":"The sacred cult revisited: the pavement of the baptistery of San Giovanni, Florence","volume":"16","author":"Williams","year":"1994","journal-title":"Math. Intell."}],"container-title":["Heliyon"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S2405844022001517?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S2405844022001517?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T10:47:08Z","timestamp":1762512428000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S2405844022001517"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["S2405844022001517"],"URL":"https:\/\/doi.org\/10.1016\/j.heliyon.2022.e08863","relation":{},"ISSN":["2405-8440"],"issn-type":[{"value":"2405-8440","type":"print"}],"subject":[],"published":{"date-parts":[[2022,2]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"The bi-periodic Horadam sequence and some perturbed tridiagonal 2-Toeplitz matrices: A unified approach","name":"articletitle","label":"Article Title"},{"value":"Heliyon","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.heliyon.2022.e08863","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2022 The Author(s). Published by Elsevier Ltd.","name":"copyright","label":"Copyright"}],"article-number":"e08863"}}