Weighted sum of the sixth powers of Horadam numbers

Kunle Adegoke, Chiachen Hsu and Olawanle Layeni
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 280–288
DOI: 10.7546/nntdm.2025.31.2.280-288
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Authors and affiliations

Kunle Adegoke
Department of Physics and Engineering Physics, Obafemi Awolowo University
220005, Ile-Ife, Nigeria

Chiachen Hsu
No. 605, Daxue S. Rd., Nanzi District,
Kaohsiung City, Taiwan

Olawanle Layeni
Department of Mathematics, Obafemi Awolowo University
220005, Ile-Ife, Nigeria

Abstract

Ohtsuka and Nakamura found simple formulas for \sum_{j=1}^n{F_j^6} and \sum_{j=1}^n{L_j^6}, where F_k and L_k are the k-th Fibonacci and Lucas numbers, respectively. In this note we extend their results to a general second order sequence by deriving a formula for \sum_{j=1}^n{(-1/q^3)^jw_{j + t}^6}, where (w_j(w_0,w_1;p,q)) is the Horadam sequence defined by w_0,\,w_1;\,w_j = pw_{j - 1} - qw_{j - 2}\, (j \ge 2); where t is an arbitrary integer and w_0, w_1, p and q are arbitrary complex numbers, with p\ne 0 and q\ne 0. As a by-product we establish a divisibility property for the generalized Fibonacci sequence.

Keywords

  • Fibonacci number
  • Lucas number
  • Horadam sequence
  • Summation identity
  • Sixth power
  • Divisibility property.

2020 Mathematics Subject Classification

  • 11B39
  • 11B37

References

  1. Adegoke, K. (2018). Weighted sums of some second-order sequences. The Fibonacci Quarterly, 56(3), 252–262.
  2. Adegoke, K. (2024). Fibonacci identities via Fibonacci functions. Journal of Integer Sequences, 27, Article 24.6.2.
  3. Benjamin, A. T. & Quinn, J. J. (2003). Proofs that Really Count: The Art of Combinatorial Proof. The Mathematical Association of America.
  4. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161–176.
  5. Larcombe, P. J. (2017). Horadam sequences: A survey update and extension. Bulletin of the Institute of Combinatorics and its Applications, 80, 99–118.
  6. Ohtsuka, H. & Nakamura, S. (2010). A new formula for the sum of the sixth powers of Fibonacci numbers. Congressus Numerantium. Proceedings of the thirteenth conference on Fibonacci numbers and their applications, 201, 297–300.
  7. Ribenboim, P. (2000). My Numbers, My Friends. Springer, New York.

Manuscript history

  • Received: 23 November 2024
  • Accepted: 7 May 2025
  • Online First: 9 May 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Adegoke, K., Hsu, C., & Layeni, O. (2025). Weighted sum of the sixth powers of Horadam numbers. Notes on Number Theory and Discrete Mathematics, 31(2), 280-288, DOI: 10.7546/nntdm.2025.31.2.280-288.

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