**Adnan Karataş**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 3, Pages 458–465

DOI: 10.7546/nntdm.2022.28.3.458-465

**Full paper (PDF, 149 Kb)**

## Details

### Authors and affiliations

Adnan Karataş

*Department of Mathematics, Faculty of Arts and Sciences, Pamukkale University
Denizli, Turkey
*

### Abstract

In this study, we introduce the complex Leonardo numbers and give some of their properties including Binet formula, generating function, Cassini and d’Ocagne’s identities. Also, we calculate summation formulas for complex Leonardo numbers involving complex Fibonacci and Lucas numbers.

### Keywords

- Complex numbers
- Fibonacci sequence
- Lucas sequence

### 2020 Mathematics Subject Classification

- 15A66
- 11B37
- 11Y55

### References

- Akhtaruzzaman, M., & Shafie, A. A. (2011). Geometrical substantiation of
*phi*, the golden ratio and the baroque of nature, architecture, design and engineering.*International Journal of Arts*, 1 (1), 1–22. - Alp, Y., & Koçer, E. G. (2021). Hybrid Leonardo numbers.
*Chaos, Solitons & Fractals*, 150, Article 111128. - Alp, Y., & Koçer, E. G. (2021). Some properties of Leonardo numbers.
*Konuralp Journal of Mathematics (KJM)*, 9(1), 183–189. - Catarino, P. (2015). A note on
*h*(*x*)-Fibonacci quaternion polynomials.*Chaos, Solitons & Fractals*, 77, 1–5. - Catarino, P., & Borges, A. (2019). On Leonardo numbers.
*Acta Mathematica Universitatis Comenianae*, 89(1), 75–86. - Flaut, C., & Savin, D. (2015). Quaternion algebras and generalized Fibonacci–Lucas quaternions.
*Advances in Applied Clifford Algebras*, 25(4), 853–862. - Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions.
*The American Mathematical Monthly*, 70(3), 289–291. - Karataş, A., & Halıcı, S. (2017). Horadam octonions.
*Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica*, 25(3), 97–106. - Kazlacheva, Z. (2017). An investigation of application of the golden ratio and Fibonacci sequence in fashion design and pattern making.
*IOP Conference Series: Materials Science and Engineering*, 254(17), Article 172013. - Koshy, T. (2011). Fibonacci, Lucas, and Pell numbers, and Pascal’s triangle.
*Mathematical Spectrum*, 43(3), 125–132. - Koshy, T. (2019).
*Fibonacci and Lucas Numbers with Applications*. John Wiley & Sons. -
Kürüz, F., Dağdeviren, A., & Catarino, P. (2021). On Leonardo Pisano hybrinomials.
*Mathematics*, 9(22), Article 2923. - Shannon, A. G. (2019). A note on generalized Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 25(3), 97–101.

### Manuscript history

- Received: 30 March 2022
- Revised: 26 July 2022
- Accepted: 28 July 2022
- Online First: 29 July 2022

## Related papers

- Shannon, A. G. (2019). A note on generalized Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 25(3), 97–101. - Shattuck, M. (2022). Combinatorial proofs of identities for the generalized Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 28(4), 778-790.

## Cite this paper

Karataş, A. (2022). On complex Leonardo numbers. *Notes on Number Theory and Discrete Mathematics*, 28(3), 458-465, DOI: 10.7546/nntdm.2022.28.3.458-465.