{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T01:12:26Z","timestamp":1770772346453,"version":"3.50.0"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T00:00:00Z","timestamp":1741564800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Federal University of Tocantins"},{"name":"Research Centre CMAT-UTAD (Polo of Research Centre CMAT\u2014Centre of Mathematics of University of Minho)"},{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"name":"Mathematics Centre of the University of Minho (CMAT\/UM)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study introduces an innovative approach to Mersenne-type numbers. This paper introduces a new class of numbers, which we call Gersenne numbers. The aim of this paper is to define the Gersenne sequence and to investigate some of their properties, such as the recurrence relation, the summation formula, and the generating function. Moreover, the classical identities are derived, such as the Tagiuri\u2013Vajda, Catalan, Cassini, and d\u2019Ocagne identities for Gersenne numbers.<\/jats:p>","DOI":"10.3390\/axioms14030203","type":"journal-article","created":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T06:59:59Z","timestamp":1741589999000},"page":"203","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5221-6087","authenticated-orcid":false,"given":"Douglas Catulio","family":"Santos","sequence":"first","affiliation":[{"name":"Education Department of the State of Bahia, Barreiras 41745-004, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6684-9961","authenticated-orcid":false,"given":"Eudes Antonio","family":"Costa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Federal University of Tocantins, Arraias 77330-000, Brazil"},{"name":"Department of Mathematics, University of Tr\u00e1s-os-Montes and Alto Douro, 5000-801 Vila Real, Portugal"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,10]]},"reference":[{"key":"ref_1","unstructured":"Sloane, N.J.A. (2025, February 07). The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation Inc.. Available online: https:\/\/oeis.org."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Deza, E. (2021). Mersenne Numbers and Fermat Numbers, World Scientific.","DOI":"10.1142\/12100"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1112\/jlms\/s1-28.3.285","article-title":"A Remark on the Primeness of Mersenne Numbers","volume":"1","author":"Hall","year":"1953","journal-title":"J. Lond. Math. Soc."},{"key":"ref_4","unstructured":"Delello, B.A. (1986). Perfect Numbers and Mersenne Primes. 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Math."},{"key":"ref_13","first-page":"147","article-title":"Generalization of Gaussian Mersenne numbers and their new families","volume":"1","author":"Kumari","year":"2025","journal-title":"Bull. Transilv. Univ. Bras."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"244","DOI":"10.55016\/ojs\/cdm.v18i2.74415","article-title":"On some new families of k-Mersenne and generalized k-Gaussian Mersenne numbers and their polynomials","volume":"18","author":"Kumari","year":"2023","journal-title":"Contrib. Discret. Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Benjamin, A.T., and Quinn, J.J. (2003). Proofs That Really Count: The Art of Combinatorial Proof MAA, American Mathematical Society.","DOI":"10.5948\/9781614442080"},{"key":"ref_16","first-page":"20","article-title":"Uma f\u00f3rmula de tipo Binet para os n\u00fameros de Geonardo","volume":"201","author":"Beites","year":"2023","journal-title":"Gaz. 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