The metallic mean (also metallic ratio, metallic constant, or noble mean) of a natural number n is a positive real number, denoted here ${\displaystyle S_{n},}$ that satisfies the following equivalent characterizations:

Metallic means are (successive) derivations of the golden (${\displaystyle n=1}$) and silver ratios (${\displaystyle n=2}$), and share some of their interesting properties. The term "bronze ratio" (${\displaystyle n=3}$) (Cf. Golden Age and Olympic Medals) and even metals such as copper (${\displaystyle n=4}$) and nickel (${\displaystyle n=5}$) are occasionally found in the literature.

In terms of algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than ${\displaystyle 1}$ and have ${\displaystyle -1}$ as their norm.

The defining equation ${\displaystyle x^{2}-nx-1=0}$ of the nth metallic mean is the characteristic equation of a linear recurrence relation of the form ${\displaystyle x_{k}=nx_{k-1}+x_{k-2}.}$ It follows that, given such a recurrence the solution can be expressed as

where ${\displaystyle S_{n}}$ is the nth metallic mean, and a and b are constants depending only on ${\displaystyle x_{0}}$ and ${\displaystyle x_{1}.}$ 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 ${\displaystyle n=1,}$ ${\displaystyle S_{n}}$ is the golden ratio. If ${\displaystyle x_{0}=0}$ and ${\displaystyle x_{1}=1,}$ the sequence is the Fibonacci sequence, and the above formula is Binet's formula. If ${\displaystyle n=1,x_{0}=2,x_{1}=1}$ one has the Lucas numbers. If ${\displaystyle n=2,}$ the metallic mean is called the silver ratio, and the elements of the sequence starting with ${\displaystyle x_{0}=0}$ and ${\displaystyle x_{1}=1}$ are called the Pell numbers.

The defining equation ${\textstyle x=n+{\frac {1}{x}}}$ of the nth metallic mean induces the following geometrical interpretation.

Consider a rectangle such that the ratio of its length L to its width W is the nth metallic ratio. If one remove from this rectangle n squares of side length W, one gets a rectangle similar to the original rectangle; that is, a rectangle with the same ratio of the length to the width (see figures).