Fibonacci Numbers as Sums of Consecutive Terms in $k$-Generalized Fibonacci Sequence
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912178586517504 |
|---|---|
| author | Alvarenga, Roberto Chaves, Ana Paula Ramos, Maria Eduarda Silva, Matheus Sosa, Marcos |
| author_facet | Alvarenga, Roberto Chaves, Ana Paula Ramos, Maria Eduarda Silva, Matheus Sosa, Marcos |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fibonacci Numbers as Sums of Consecutive Terms in $k$-Generalized Fibonacci Sequence Alvarenga, Roberto Chaves, Ana Paula Ramos, Maria Eduarda Silva, Matheus Sosa, Marcos Number Theory 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. |
| title | Fibonacci Numbers as Sums of Consecutive Terms in $k$-Generalized Fibonacci Sequence |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.03438 |