Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence

Elis Gardel Costa Mesquista; Paula Maria Machado Cruz Catarino; Eudes Antonio Costa

doi:10.33434/cams.1754577

EN

Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence

Abstract

The focus of this paper is to study the $2^k$–Fibonacci 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.

Keywords

Binet’s formula , Fibonacci-type sequences , $2^k$-Fibonacci sequence , Generating function

References

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Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Elis Gardel Costa Mesquista
0000-0003-2385-4108
Brazil

Paula Maria Machado Cruz Catarino
0000-0001-6917-5093
Portugal

Eudes Antonio Costa ^*
0000-0001-6684-9961
Brazil

Early Pub Date

December 5, 2025

Publication Date

December 8, 2025

Submission Date

September 2, 2025

Acceptance Date

December 1, 2025

Published in Issue

Year 2025 Volume: 8 Number: 4

DOI

https://doi.org/10.33434/cams.1754577

IZ

https://izlik.org/JA48RL74HX

APA

Costa Mesquista, E. G., Catarino, P. M. M. C., & Costa, E. A. (2025). Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence. Communications in Advanced Mathematical Sciences, 8(4), 225-246. https://doi.org/10.33434/cams.1754577

AMA

1.Costa Mesquista EG, Catarino PMMC, Costa EA. Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence. Communications in Advanced Mathematical Sciences. 2025;8(4):225-246. doi:10.33434/cams.1754577

Chicago

Costa Mesquista, Elis Gardel, Paula Maria Machado Cruz Catarino, and Eudes Antonio Costa. 2025. “Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence”. Communications in Advanced Mathematical Sciences 8 (4): 225-46. https://doi.org/10.33434/cams.1754577.

EndNote

Costa Mesquista EG, Catarino PMMC, Costa EA (December 1, 2025) Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence. Communications in Advanced Mathematical Sciences 8 4 225–246.

IEEE

[1]E. G. Costa Mesquista, P. M. M. C. Catarino, and E. A. Costa, “Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence”, Communications in Advanced Mathematical Sciences, vol. 8, no. 4, pp. 225–246, Dec. 2025, doi: 10.33434/cams.1754577.

ISNAD

Costa Mesquista, Elis Gardel - Catarino, Paula Maria Machado Cruz - Costa, Eudes Antonio. “Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence”. Communications in Advanced Mathematical Sciences 8/4 (December 1, 2025): 225-246. https://doi.org/10.33434/cams.1754577.

JAMA

1.Costa Mesquista EG, Catarino PMMC, Costa EA. Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence. Communications in Advanced Mathematical Sciences. 2025;8:225–246.

MLA

Costa Mesquista, Elis Gardel, et al. “Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence”. Communications in Advanced Mathematical Sciences, vol. 8, no. 4, Dec. 2025, pp. 225-46, doi:10.33434/cams.1754577.

Vancouver

1.Elis Gardel Costa Mesquista, Paula Maria Machado Cruz Catarino, Eudes Antonio Costa. Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence. Communications in Advanced Mathematical Sciences. 2025 Dec. 1;8(4):225-46. doi:10.33434/cams.1754577

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Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence

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Exploring Generalized $2^k$-Fibonacci Sequence: A New Family of the Fibonacci Sequence

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