On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases

Munesh Kumari; Kalika Prasad; Mritunjay Kumar Singh

doi:10.32323/ujma.1854795

On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases

Abstract

In this article, we discuss three new extensions of the Leonardo numbers in a generalized way, which we call the generalized Silver, Bronze, and Copper Leonardo numbers that converge to the Silver, Bronze, and Copper ratios, unifying existing metallic Leonardo sequences. We investigate their fundamental algebraic properties, including recurrence relations, limiting ratios, Binet-type formulas, explicit expressions, and partial sums. Furthermore, we explore their ordinary and exponential generating functions. Finally, we illuminate the inherent relationships between these generalized metallic Leonardo numbers and the well-established metallic Fibonacci and Lucas numbers, revealing their interconnectedness within a broader mathematical structure.

Keywords

Silver Leonardo numbers, Bronze Leonardo numbers, Copper Leonardo numbers, Limiting ratio, Generating functions, Metallic sequences

References

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Details

Primary Language

English

Subjects

Numerical and Computational Mathematics (Other), Algebra and Number Theory

Journal Section

Research Article

Authors

Munesh Kumari
0000-0002-6541-0284
India

Kalika Prasad ^*
0000-0002-3653-5854
India

Mritunjay Kumar Singh
0000-0002-7427-0076
India

Early Pub Date

March 17, 2026

Publication Date

March 17, 2026

Submission Date

January 10, 2026

Acceptance Date

March 15, 2026

Published in Issue

Year 2026 Volume: 9 Number: 1

DOI

https://doi.org/10.32323/ujma.1854795

IZ

https://izlik.org/JA63DH46GD

APA

Kumari, M., Prasad, K., & Singh, M. K. (2026). On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases. Universal Journal of Mathematics and Applications, 9(1), 44-52. https://doi.org/10.32323/ujma.1854795

AMA

1.Kumari M, Prasad K, Singh MK. On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases. Univ. J. Math. Appl. 2026;9(1):44-52. doi:10.32323/ujma.1854795

Chicago

Kumari, Munesh, Kalika Prasad, and Mritunjay Kumar Singh. 2026. “On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases”. Universal Journal of Mathematics and Applications 9 (1): 44-52. https://doi.org/10.32323/ujma.1854795.

EndNote

Kumari M, Prasad K, Singh MK (March 1, 2026) On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases. Universal Journal of Mathematics and Applications 9 1 44–52.

IEEE

[1]M. Kumari, K. Prasad, and M. K. Singh, “On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases”, Univ. J. Math. Appl., vol. 9, no. 1, pp. 44–52, Mar. 2026, doi: 10.32323/ujma.1854795.

ISNAD

Kumari, Munesh - Prasad, Kalika - Singh, Mritunjay Kumar. “On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases”. Universal Journal of Mathematics and Applications 9/1 (March 1, 2026): 44-52. https://doi.org/10.32323/ujma.1854795.

JAMA

1.Kumari M, Prasad K, Singh MK. On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases. Univ. J. Math. Appl. 2026;9:44–52.

MLA

Kumari, Munesh, et al. “On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases”. Universal Journal of Mathematics and Applications, vol. 9, no. 1, Mar. 2026, pp. 44-52, doi:10.32323/ujma.1854795.

Vancouver

1.Munesh Kumari, Kalika Prasad, Mritunjay Kumar Singh. On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases. Univ. J. Math. Appl. 2026 Mar. 1;9(1):44-52. doi:10.32323/ujma.1854795