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

Year 2025, Volume: 8 Issue: 4, 225 - 246, 08.12.2025

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

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

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

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