Metallic means

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Metallic means (metallic ratios), also called silver means (silver ratios), are a generalization of the golden ratio.

Metallic means (metallic ratios)

n	n + 2√  n 2 + 4 2	Value	A-number	Name
0	(0 + 2√  4 ) / 2	1
1	(1 + 2√  5 ) / 2	1.618033989	A001622	Golden ratio
2	(2 + 2√  8 ) / 2	2.414213562	A014176	Silver ratio
3	(3 + 2√  13 ) / 2	3.302775638	A098316
4	(4 + 2√  20 ) / 2	4.236067978	A098317
5	(5 + 2√  29 ) / 2	5.192582404	A098318
6	(6 + 2√  40 ) / 2	6.162277660	A176398
7	(7 + 2√  53 ) / 2	7.140054945	A176439
8	(8 + 2√  68 ) / 2	8.123105626	A176458
9	(9 + 2√  85 ) / 2	9.109772229	A176522

The more general simple continued fraction expressions

$n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{\ddots }}}}}}}}}}=[n;n,n,n,n,\dots ]={\frac {1}{2}}\left(n+{\sqrt {n^{2}+4}}\right)\,$

are known as the silver means or metallic means of the successive natural numbers. The golden ratio is the silver mean between 

1

and 

2

, while the silver ratio is the silver mean between 

2

and 

3

. The term "bronze ratio", or terms using other names of metals, are occasionally used to name subsequent silver means. The values of the first ten silver means are shown at right. Notice that each silver mean is a root of the simple quadratic equation

$x^{2}-nx-1=0,\,$

where 

n

is any positive natural number. Hence

$x-n={\frac {1}{x}}={\frac {1}{n+(x-n)}},\,$

which leads to the above continued fraction.

The silver mean of 

n

is also given by the integral

$S_{n}=\int _{0}^{n}{\left({x \over {2{\sqrt {x^{2}+4}}}}+{{n+2} \over {2n}}\right)}\,{\rm {d}}x.$