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

TL;DR

This work analyzes when sums of consecutive terms of a $k$-generalized Fibonacci sequence equal a standard Fibonacci number. It builds a Binet-type expression from the roots of the characteristic polynomial and applies Baker’s theory on linear forms in logarithms to derive finiteness results. The main result proves that for fixed $d$ and $k\ge 3$, the equation $F_n^{(k)}+\cdots+F_{n+d}^{(k)}=F_m$ has only finitely many solutions, implying a finite intersection between the two sequences; the case $k=2$ is treated separately, with corollaries for certain $d$. The approach yields explicit bounds and is illustrated with concrete examples, highlighting the Diophantine nature of these sequence intersections.

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.

Theorems & Definitions (18)

• Proposition 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6