The metallic mean (also metallic ratio, metallic constant, or noble means[1]) of a natural number n is a positive real number, denoted here that satisfies the following equivalent characterizations:
Metallic means are generalizations of the golden ratio () and silver ratio (), and share some of their interesting properties. The term "bronze ratio" (), and terms using other metals names (such as copper or nickel), are occasionally used to name subsequent metallic means.[2] [3]
In terms of algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than and have as their norm.
The defining equation of the nth metallic mean is the characteristic equation of a linear recurrence relation of the form It follows that, given such a recurrence the solution can be expressed as
where is the nth metallic mean, and a and b are constants depending only on and Since the inverse of a metallic mean is less than 1, this formula implies that the quotient of two consecutive elements of such a sequence tends to the metallic mean, when k tends to the infinity.
For example, if is the golden ratio. If and the sequence is the Fibonacci sequence, and the above formula is Binet's formula. If one has the Lucas numbers. If the metallic mean is called the silver ratio, and the elements of the sequence starting with and are called the Pell numbers. The third metallic mean is sometimes called the "bronze ratio".