Fibonacci Numbers as Sums of Consecutive Terms in $k$-Generalized Fibonacci Sequence

Abstract

Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms. This article is devoted to investigating when the sum of consecutive numbers in the k-generalized Fibonacci sequence belongs to the Fibonacci sequence. Namely, given d,k \in \N, with k \geq 3, our main theorem states that there are at most finitely many n \in \N such that F_n^{(k)} + \cdots + F_{n+d}^{(k)} is a Fibonacci number. In particular, the intersection between the Fibonacci sequence and the k-generalized Fibonacci sequence is finite.