Unrestricted Lichtenberg Hybrid Sequences

Year 2025, Volume: 13 Issue: 3, 156 - 164, 06.09.2025

Gamaliel Morales

https://doi.org/10.36753/mathenot.1689222

Cited By: 1

Abstract

This research introduces a novel category of hybrid numbers, with components represented by unrestricted Lichtenberg numbers. We present some recurrence relations, summation formulas, the Binet formula, and the generating function associated with these numbers. Additionally, a comprehensive Vajda's identity is derived, which encompasses Catalan's, Cassini's and d'Ocagne's identities as specific cases.

Keywords

Binet formula , Hybrid numbers , Lichtenberg sequence , Mersenne sequence , Unrestricted sequences

References

• [1] Hinz, A. M.: The Lichtenberg sequence. The Fibonacci Quarterly. 55(2), 2-12 (2017).
• [2] Stockmeyer, P. K.: An exploration of sequence A000975. The Fibonacci Quarterly. 55(5), 174-185 (2017).
• [3] Horadam, A. F.: Jacobsthal representation numbers. The Fibonacci Quarterly. 43(1), 40-54 (1996).
• [4] Catarino, P., Campos, H., Vasco, P.: On the Mersenne sequence. Annales Mathematicae et Informaticae. 46, 37-53 (2016).
• [5] Özdemir, M.: Introduction to hybrid numbers. Advances in Applied Clifford Algebras. 28, 1-32 (2018).
• [6] Morales, G.: Investigation of generalized Fibonacci hybrid numbers and their properties. Applied Mathematics E-Notes. 21, 110-118 (2021).
• [7] Polatlı, E.: Hybrid numbers with Fibonacci and Lucas hybrid number coefficients. Universal Journal of Mathematics and Applications. 6(3), 106-113 (2023).
• [8] Şentürk, T. D., Bilgici, G., Daşdemir, A., Ünal, Z.: A study on Horadam hybrid numbers. Turkish Journal of Mathematics. 44(4), 1212-1221 (2020).
• [9] Szynal-Liana, A., Włoch, I.: The Fibonacci hybrid numbers. Utilitas Mathematica. 110, 3-10 (2019).
• [10] Polatlı, E.: A note on ratios of Fibonacci hybrid and Lucas hybrid numbers. Notes on Number Theory and Discrete Mathematics. 27(3), 73-78 (2021).
• [11] Szynal-Liana, A., Włoch, I.: On Pell and Pell-Lucas hybrid numbers. Commentationes Mathematicae. 58(1), 11-17 (2018).
• [12] Szynal-Liana, A., Włoch, I.: On Jacobsthal and Jacobsthal-Lucas hybrid numbers. Annales Mathematicae Silesianae. 33, 276-283 (2019).
• [13] Szynal-Liana, A., Włoch, I.: On generalized Mersenne hybrid numbers. Annales Universitatis Mariae Curie- Sklodowska, sectio A–Mathematica. 74(1), 77-84 (2020).
• [14] Taşci, D., Sevgi, E.: Some properties between Mersenne, Jacobsthal and Jacobsthal-Lucas hybrid numbers, Chaos, Solitons & Fractals. 146, 1-7 (2021).
• [15] Özkan, E., Uysal, M.: Mersenne-Lucas hybrid numbers. Mathematica Montisnigri. 52, 17-29 (2021).
• [16] Morales, G.: On the Lichtenberg hybrid quaternions. Mathematica Moravica. 29(1), 31-41 (2025).
• [17] Morales, G.: Properties between Lichtenberg, Jacobsthal and Mersenne Gaussian numbers. Konuralp Journal of Mathematics. 13(1), 21-27 (2025).
• [18] Maheswari, D., Devi B. M., Devibala, S.: Unrestricted Mersenne and Mersenne-Lucas hybrid sequences. Indian Journal of Science and Technology. 16(41), 3591-3598 (2023).