A note on generalized and extended Leonardo sequences

Anthony G. Shannon and Ömür Deveci
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 1, Pages 109—114
DOI: 10.7546/nntdm.2022.28.1.109-114
Download PDF (605 Kb)


Authors and affiliations

Anthony G. Shannon
Warrane College, the University of New South Wales
Kensington, NSW 2033, Australia

Ömür Deveci
Department of Mathematics, Faculty of Science and Letters
Kafkas University, 36100 Kars, Turkey


This note considers some real and complex extensions and generalizations of the Leonardo sequence, which is embedded within each of these two types of intriguing sequences, intriguing because there are still some unanswered questions. The connections between inhomogeneous and homogeneous forms are used as examples of a possible reason that the Leonardo sequences have been, in a sense, historically neglected.


  • Fibonacci sequence
  • Lucas sequence
  • Leonardo sequence
  • Inhomogeneous recurrence relations

2020 Mathematics Subject Classification

  • 11B37
  • 11B39


  1. Asveld, P. R. J. (1987). A family of Fibonacci-like sequences. The Fibonacci Quarterly, 25(1), 81–83.
  2. Catarino, P., & Borges, A. (2019). On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89(1), 75–86.
  3. Deveci, Ö. (2015). The Pell–Padovan sequences and the Jacobsthal–Padovan sequences in finite groups. Utilitas Mathematica, 98, 257–270.
  4. Dijkstra, E. W. (1981). Fibonacci numbers and Leonardo numbers. EWD797. Available online at: https://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD797.html
  5. Horadam, A. F., & Shannon, A. G. (1988). Asveld’s polynomials. In A.N. Philippou, A. F. Horadam and G. E. Bergum (Eds.). Applications of Fibonacci Numbers, Volume 2. (pp. 163–176). Dordrecht: Kluwer.
  6. Jarden, D. (1966). Recurring Sequences. Jerusalem: Riveon Lematematika, 95–102.
  7. King, C. (1963). Leonardo Fibonacci. The Fibonacci Quarterly, 1(4), 15–19.
  8. Lucas, E. (1878). Théorie des fonctions numériques simplement périodiques. American Journal of Mathematics, 1, 184–240, 289–321.
  9. Riordan, J. (1954). Discordant permutations. Scripta Mathematica, 20, 14–23.
  10. Shannon, A. G. (2019). A note on generalized Leonardo numbers. Notes on Number Theory and Discrete Mathematics, 25(3), 97–101.
  11. Sloane, N. J. A., & Plouffe, S. (1995). The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, Available online at: https://oeis.org.

Manuscript history

  • Received: 28 December 2021
  • Accepted: 11 February 2022
  • Online First: 21 February 2022

Related papers

Cite this paper

Shannon, A. G., & Deveci, Ö. (2022). A note on generalized and extended Leonardo sequences. Notes on Number Theory and Discrete Mathematics, 28(1), 109-114, DOI: 10.7546/nntdm.2022.28.1.109-114.

Comments are closed.