Metallic mean

The metallic mean (also metallic ratio, metallic constant, or noble mean[1]) of a natural number n is a positive real number, denoted here

that satisfies the following equivalent characterizations: Metallic means are (successive) derivations of the golden (

Golden Age and Olympic Medals) and even metals such as copper (

[2] [3] In terms of algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than

of the nth metallic mean is the characteristic equation of a linear recurrence relation of the form

is the nth metallic mean, and a and b are constants depending only on

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.

the metallic mean is called the silver ratio, and the elements of the sequence starting with

of the nth metallic mean induces the following geometrical interpretation.

Some metallic means appear as segments in the figure formed by a regular polygon and its diagonals.

are defined recursively by the initial conditions K0 = 0 and K1 = 1, and the recurrence relation Proof: The equality is immediately true for

One has also [citation needed] The odd powers of a metallic mean are themselves metallic means.

More precisely, if n is an odd natural number, then

The definition of metallic means implies that

is an integer that satisfies the given recurrence relation.

This results from the identity This completes the proof, given that the initial values are easy to verify.

In particular, one has and, in general,[citation needed] where For even powers, things are more complicated.

If n is a positive even integer then[citation needed] Additionally,[citation needed] For the square of a metallic ratio we have:

of a negative integer −n as the positive solution of the equation

The metallic mean of −n is the multiplicative inverse of the metallic mean of n: Another generalization consists of changing the defining equation from

If is any root of the equation, one has The silver mean of m is also given by the integral[citation needed] Another form of the metallic mean is[citation needed] A tangent half-angle formula gives

is a positive integer, as it is with some primitive Pythagorean triangles.

Metallic means are precisely represented by some primitive Pythagorean triples, a2 + b2 = c2, with positive integers a < b < c. In a primitive Pythagorean triple, if the difference between hypotenuse c and longer leg b is 1, 2 or 8, such Pythagorean triple accurately represents one particular metallic mean.

The cotangent of the quarter of smaller acute angle of such Pythagorean triangle equals the precise value of one particular metallic mean.

Such Pythagorean triangle (a, b, c) yields the precise value of a particular metallic mean

where α is the smaller acute angle of the Pythagorean triangle and the metallic mean index is

For example, the primitive Pythagorean triple 20-21-29 incorporates the 5th metallic mean.

Cotangent of the quarter of smaller acute angle of the 20-21-29 Pythagorean triangle yields the precise value of the 5th metallic mean.

Similarly, the Pythagorean triangle 3-4-5 represents the 6th metallic mean.

Likewise, the Pythagorean triple 12-35-37 gives the 12th metallic mean, the Pythagorean triple 52-165-173 yields the 13th metallic mean, and so on.