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

TL;DR

This paper investigates the conditions under which sums of consecutive terms in the k-generalized Fibonacci sequence are Fibonacci numbers, establishing that such occurrences are finite and the intersection between the two sequences is limited.

Contribution

It proves that only finitely many sums of consecutive k-generalized Fibonacci terms are Fibonacci numbers, showing the intersection between these sequences is finite.

Findings

Finitely many sums of consecutive k-generalized Fibonacci terms are Fibonacci numbers.

The intersection between the Fibonacci sequence and the k-generalized Fibonacci sequence is finite.

Main theorem establishes bounds on when such sums can be Fibonacci numbers.

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.