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

2025/01/06 by Alvarenga, Roberto, Chaves, Ana Paula, Ramos, Maria Eduarda +2
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.03438

Abstract

Let (F_n^(k))_{n≥ -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, …, 0, 1\), 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 ∈ \N, with k ≥ 3, our main theorem states that there are at most finitely many n ∈ \N such that F_n^(k) + ⋯ + F_n+d^(k) is a Fibonacci number. In particular, the intersection between the Fibonacci sequence and the k-generalized Fibonacci sequence is finite.

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