{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T08:58:33Z","timestamp":1767085113447,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T00:00:00Z","timestamp":1735257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"PROPESQ-UFT","award":["UIDB\/00013\/2020","UIDP\/00013\/2020"],"award-info":[{"award-number":["UIDB\/00013\/2020","UIDP\/00013\/2020"]}]},{"name":"Portuguese Funds through FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UIDB\/00013\/2020","UIDP\/00013\/2020"],"award-info":[{"award-number":["UIDB\/00013\/2020","UIDP\/00013\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we introduce a new family of sequences related to Horadam-type sequences. Specifically, we consider the repunit sequence {rn}n\u22650, which is defined by the initial terms r0=0 and r1=1 and follows the Horadam recurrence relation given by rn=11rn\u22121\u221210rn\u22122 for n\u22652. Many studies have explored generalizations of integer sequences in different directions: some by preserving the initial terms, some by preserving the recurrence relation, and some by considering different numerical sets beyond positive integers. In this article, we take the third approach. Specifically, we study these sequences in the context of the tricomplex ring T. We define the Tricomplex Repunit sequence {trn}n\u22650, with initial terms tr0=(0,1,11) and tr1=(1,11,111), and governed by the recurrence relation trn=11trn\u22121\u221210trn\u22122, for n\u22652. This sequence is also a Horadam-type sequence but defined in the tricomplex ring T. In this paper, we establish the properties of the Tricomplex Repunit sequence and establish several new as well as well-known identities associated with it, including Binet\u2019s formula, Tagiuri\u2013Vajda\u2019s identity, d\u2019Ocagne\u2019s identity, and Catalan\u2019s identity. We also derive the generating function for this sequence. Furthermore, we study various additional properties of these generalized sequences and establish results concerning the summation of terms related to the Tricomplex Repunit sequence, and one of our main goals is to determine analogous or symmetrical properties for the Tricomplex Repunit sequence to those we know for the ordinary repunit sequence.<\/jats:p>","DOI":"10.3390\/sym17010028","type":"journal-article","created":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T09:13:32Z","timestamp":1735290812000},"page":"28","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6684-9961","authenticated-orcid":false,"given":"Eudes A.","family":"Costa","sequence":"first","affiliation":[{"name":"Department of Mathematics, Federal University of Tocantins, Arraias 77330-000, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6917-5093","authenticated-orcid":false,"given":"Paula M. M. C.","family":"Catarino","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Tr\u00e1s-os-Montes and Alto Douro, 5000-801 Vila Real, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5221-6087","authenticated-orcid":false,"given":"Douglas C.","family":"Santos","sequence":"additional","affiliation":[{"name":"Education Department of the State of Bahia, Barreiras 41745-004, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,27]]},"reference":[{"key":"ref_1","unstructured":"Sloane, N.J.A. (2024). The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation Inc.. Available online: https:\/\/oeis.org."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"54","DOI":"10.22481\/intermaths.v5i1.14922","article-title":"A note on Repunit number sequence","volume":"5","author":"Santos","year":"2024","journal-title":"Intermaths"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"241","DOI":"10.21167\/cqdv23n1ic2023241254","article-title":"Um passeio pela sequ\u00eancia repunidade","volume":"23","author":"Santos","year":"2023","journal-title":"CQD-Rev. Eletr\u00f4nica Paul. Matem\u00e1tica"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"K\u0131z\u0131late\u015f, C., Catarino, P., and Tu\u01e7lu, N. (2019). On the bicomplex generalized Tribonacci quaternions. Mathematics, 7.","DOI":"10.3390\/math7010080"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Costa, E., Catarino, P., Monteiro, F., Sousa, V., and Santos, D. (2024). Tricomplex Fibonacci Numbers: A New Family of Fibonacci-type Sequences. Mathematics, 12.","DOI":"10.3390\/math12233723"},{"key":"ref_6","unstructured":"Baumgart, J.K. (1992). T\u00f3picos de Hist\u00f3ria da Matem\u00e1tica Para uso em Sala de Aula: \u00c1lgebra, Atual Editora."},{"key":"ref_7","unstructured":"Boyer, C.B., and Merzbach, U.C. (2011). A History of Mathematics, John Wiley and Sons."},{"key":"ref_8","unstructured":"Felzenszwalb, B. (1979). \u00c1lgebras de Dimens\u00e3o Finitas, Instituto de Matem\u00e1tica pura e Aplicada (Col\u00f3quio Brasileiro de Matem\u00e1tica)."},{"key":"ref_9","unstructured":"Olariu, S. (2002). Complex Numbers in n Dimensions [Mathematics Studies], Elsevier Science B. V."},{"key":"ref_10","unstructured":"Olariu, S. (2000). Complex numbers in three dimensions. arXiv."},{"key":"ref_11","first-page":"26","article-title":"Tri-complex rough neutrosophic similarity measure and its application in multi-attribute decision making","volume":"11","author":"Mondal","year":"2015","journal-title":"Crit. Rev."},{"key":"ref_12","unstructured":"Ottoni, A., de Deus, N.C.L., and Ottoni, J.E.O. (2024). A \u00c1lgebra dos n\u00fameros tern\u00e1rios. Rev. Matem\u00e1tica UFOP, 1."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Richter, W. (2022). On complex numbers in higher dimensions. Axioms, 11.","DOI":"10.3390\/axioms11010022"},{"key":"ref_14","unstructured":"Beiler, A.H. (1966). Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover. [2nd ed.]."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Deza, E., and Deza, M. (2012). Figurate Numbers, World Scientific.","DOI":"10.1142\/9789814355490"},{"key":"ref_16","unstructured":"Tarasov, B.V. (2007). The Concrete Theory of Numbers: Initial Numbers and Wonderful Properties of Numbers Repunit, Cornell University. Arxiv.org [math.GM]."},{"key":"ref_17","unstructured":"Yates, S. (1992). Repunits and Repetends, Star Publishing Co., Inc."},{"key":"ref_18","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_19","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_20","unstructured":"Costa, E., Santos, D., Catarino, P., and Spreafico, E. (Rev. Matem\u00e1tica UFOP, 2024). On Gaussian and Quaternion Repunit Numbers, Rev. Matem\u00e1tica UFOP, accept."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1111\/j.1949-8594.1994.tb15641.x","article-title":"Exploring repunits","volume":"94","author":"Toumasis","year":"1994","journal-title":"Sch. Sci. Math."},{"key":"ref_22","unstructured":"Costa, E., Santos, D., Monteiro, F., and Souza, V. (2024). On the Repunit sequence at negative indices. Rev. Matem\u00e1tica UFOP, 1."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Kristyan, S. (2022). Lucas sequences and Fibonacci numbers related equations. Part i.: Differential equations and sums. AIP Conference Proceedings, AIP Publishing.","DOI":"10.1063\/5.0081313"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"14371","DOI":"10.1002\/mma.7702","article-title":"New families of Horadam numbers associated with finite operators and their applications","volume":"44","year":"2021","journal-title":"Math. Methods Appl. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:01:17Z","timestamp":1760115677000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,27]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010028"],"URL":"https:\/\/doi.org\/10.3390\/sym17010028","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,12,27]]}}}