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# Metallic means

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
 1.61803
A001622 Golden ratio
 2
 (2 + 2√  8) / 2
 2.41421
A014176 Silver ratio
 3
 (3 + 2√  13) / 2
 3.30278
A098316
 4
 (4 + 2√  20) / 2
 4.23607
A098317
 5
 (5 + 2√  29) / 2
 5.19258
A098318
 6
 (6 + 2√  40) / 2
 6.16228
A176398
 7
 (7 + 2√  53) / 2
 7.14005
A176439
 8
 (8 + 2√  68) / 2
 8.12311
A176458
 9
 (9 + 2√  85) / 2
 9.10977
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
$S_{n}=\int _{0}^{n}{\left({x \over {2{\sqrt {x^{2}+4}}}}+{{n+2} \over {2n}}\right)}\,{\rm {d}}x.$ 