univ. j. math. appl. Universal Journal of Mathematics and Applications 2619-9653 Emrah Evren KARA 10.32323/ujma.1854795 Numerical and Computational Mathematics (Other) Algebra and Number Theory Sayısal ve Hesaplamalı Matematik (Diğer) Cebir ve Sayı Teorisi On Generalized Metallic Leonardo Numbers: Silver, Bronze, and Copper Cases https://orcid.org/0000-0002-6541-0284 Kumari Munesh GOVERNMENT ENGINEERING COLLEGE BHOJPUR, BIHAR, INDIA https://orcid.org/0000-0002-3653-5854 Prasad Kalika GOVERNMENT ENGINEERING COLLEGE BHOJPUR, BIHAR, INDIA https://orcid.org/0000-0002-7427-0076 Singh Mritunjay Kumar GOVERNMENT POLYTECHNIC, NAWADA 03 17 2026 9 1 44 52 01 10 2026 03 15 2026 Copyright © 2018, Universal Journal of Mathematics and Applications 2018 Universal Journal of Mathematics and Applications

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.

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