{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,16]],"date-time":"2025-12-16T23:43:38Z","timestamp":1765928618883,"version":"3.48.0"},"reference-count":32,"publisher":"Communications in Advanced Mathematical Sciences","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,12,1]]},"abstract":"<jats:p xml:lang=\"en\">The focus of this paper is to study the $2^k$\u2013Fibonacci sequence, which is defined for all integers $2^k$, and its connections with both the Fibonacci and the Fibonacci-Lucas sequences.  Among the main results, we highlight the expression of the $2^k$-Fibonacci numbers as a linear combination of Fibonacci numbers and Fibonacci-Lucas numbers. Additionally, the paper presents several identities, such as Binet's formula, the Tagiuri-Vajda identity, d'Ocagne's identity, Catalan's identity, and the generating function. Furthermore, we explore some properties of these generalized sequences and establish formulas for sums of terms involving the $2^k$-Fibonacci numbers.<\/jats:p>","DOI":"10.33434\/cams.1754577","type":"journal-article","created":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T17:30:28Z","timestamp":1764955828000},"page":"225-246","source":"Crossref","is-referenced-by-count":0,"title":["Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence"],"prefix":"10.33434","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2385-4108","authenticated-orcid":true,"given":"Elis Gardel","family":"Costa Mesquista","sequence":"first","affiliation":[{"name":"Federal University of Tocantins"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6917-5093","authenticated-orcid":true,"given":"Paula Maria Machado Cruz","family":"Catarino","sequence":"additional","affiliation":[{"name":"University of Tr as-os-Montes and Alto Douro"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6684-9961","authenticated-orcid":true,"given":"Eudes Antonio","family":"Costa","sequence":"additional","affiliation":[{"name":"Federal University of Tocantins"}]}],"member":"19060","published-online":{"date-parts":[[2025,12,8]]},"reference":[{"unstructured":"[1] N. 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