{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T08:55:06Z","timestamp":1767084906394,"version":"3.45.0"},"reference-count":21,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T00:00:00Z","timestamp":1763596800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UID\/00013\/2025"],"award-info":[{"award-number":["UID\/00013\/2025"]}]},{"DOI":"10.13039\/100030712","name":"Federal University of Tocantins","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100030712","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study is introduced a novel generalization of repunit and one-zero numbers through the formulation of the generalized One-kZero. This sequence extends the classical families of repunit and one-zero numbers by establishing a unified framework in which the parameter (k\u22650) specifies the number of consecutive zeros separating two ones in the decimal representation. We introduce the new family of sequences, the generalized One-kZero numbers, and investigate some of their properties. The main purpose is to present a generalization for the recurrence relation of kind One-Zero numbers and determine some relations and properties. The reason that led us to this method is that the recurrence relation of One-Zero and Repunit numbers has a second-order difference equation as a specific case of the Horadam-type sequence. The Binet formula, generating function, sum formulas and many other relations will therefore be much easier to find. Also, some other identities that have not been found before in the particular case of One-Zero and Repunit sequences are also included in this study.<\/jats:p>","DOI":"10.3390\/axioms14110854","type":"journal-article","created":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T09:50:26Z","timestamp":1763718626000},"page":"854","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Generalized One-kZero Numbers"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6917-5093","authenticated-orcid":false,"given":"Paula M. M. C.","family":"Catarino","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Tr\u00e1s-os-Montes and Alto Douro, 5000-801 Vila Real, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2611-8481","authenticated-orcid":false,"given":"Grieg A.","family":"Costa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bras\u00edlia, Bras\u00edlia 70910-900, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6684-9961","authenticated-orcid":false,"given":"Eudes A.","family":"Costa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tr\u00e1s-os-Montes and Alto Douro, 5000-801 Vila Real, Portugal"},{"name":"Department of Mathematics, Federal University of Tocantins, Arraias 77330-000, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1080\/00029890.1961.11989696","article-title":"A generalized Fibonacci sequence","volume":"68","author":"Horadam","year":"1961","journal-title":"Am. Math. Mon."},{"key":"ref_2","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":"ref_3","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1080\/00150517.1967.12431271","article-title":"Special properties of the sequence Wn(a, b; p, q)","volume":"5","author":"Horadam","year":"1967","journal-title":"Fibonacci Q."},{"key":"ref_4","first-page":"3611","article-title":"On generalized Fibonacci numbers","volume":"9","author":"Bacani","year":"2015","journal-title":"Appl. Math. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Edson, M., and Yayenie, O. (2009). A New Generalization of Fibonacci Sequence & Extended Binet\u2019s Formula, Walter de Gruyter GmbH & Co. KG.","DOI":"10.1515\/INTEG.2009.051"},{"key":"ref_6","first-page":"167","article-title":"The Fibonacci numbers\u2014Exposed","volume":"76","author":"Kalman","year":"2003","journal-title":"Math. Mag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"172","DOI":"10.5539\/jmr.v8n4p172","article-title":"A General Family of Fibonacci-Type Squences","volume":"8","author":"Nanhongkai","year":"2016","journal-title":"J. Math. Res."},{"key":"ref_8","unstructured":"Sloane, N.J.A. (2025). The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation Inc.. Available online: https:\/\/oeis.org."},{"key":"ref_9","unstructured":"Beiler, A.H. (1966). Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover. [2nd ed.]."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Gupta, S.S. (2025). Repunit Numbers. Exploring the Beauty of Fascinating Numbers, Springer. 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Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, Oxford University Press."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Catarino, P.M.M.C., Santos, D.C., and Costa, E.A. (2025). On t-Dimensional Gersenne Sequences and Their Symmetry Properties. Symmetry, 17.","DOI":"10.3390\/sym17071079"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Santos, D.C., Costa, E.A., and Catarino, P.M.M.C. (2025). On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence. Axioms, 14.","DOI":"10.3390\/axioms14030203"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Deza, E. (2021). Mersenne Numbers and Fermat Numbers, World Scientific.","DOI":"10.1142\/12100"},{"key":"ref_19","first-page":"6605","article-title":"On Generalized Mersenne Numbers and Extended Fermat Numbers","volume":"18","author":"Melhem","year":"2025","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_20","first-page":"69","article-title":"On generalized Mersenne numbers, their interpretations and matrix generators","volume":"72","author":"Ochalik","year":"2018","journal-title":"Ann. Univ. Mariae Curie Sk\u0142odowska Sect. A Math."},{"key":"ref_21","first-page":"90","article-title":"A study on generalized Mersenne numbers","volume":"18","author":"Soykan","year":"2021","journal-title":"J. Progress. Res. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/11\/854\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,22]],"date-time":"2025-11-22T05:28:17Z","timestamp":1763789297000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/11\/854"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,20]]},"references-count":21,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["axioms14110854"],"URL":"https:\/\/doi.org\/10.3390\/axioms14110854","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,11,20]]}}}