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

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.