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

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.