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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

${\displaystyle n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{\ddots }}}}}}}}}}=[n;n,n,n,n,\dots ]={\frac {1}{2}}\left(n+{\sqrt {n^{2}+4}}\right)\,}$
are known as the silver means or metallic means[1] 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.[1] The values of the first ten silver means are shown at right.[2] Notice that each silver mean is a root of the simple quadratic equation
${\displaystyle x^{2}-nx-1=0,\,}$
where
 n
is any positive natural number. Hence
${\displaystyle x-n={\frac {1}{x}}={\frac {1}{n+(x-n)}},\,}$

which leads to the above continued fraction.

The silver mean of
 n
is also given by the integral
${\displaystyle S_{n}=\int _{0}^{n}{\left({x \over {2{\sqrt {x^{2}+4}}}}+{{n+2} \over {2n}}\right)}\,{\rm {d}}x.}$

Notes

1. Weisstein, Eric W., Silver Ratio, from MathWorld—A Wolfram Web Resource.