貴金屬比例

貴金屬比例、貴金屬分割（英語：metallic ratio）定义为

{\frac {n+{\sqrt {n^{2}+4}}}{2}}:1

（n为自然数）

所表示的比率。

随 $n$ 值的不同，又称为第 $n$ 貴金屬比例、第 $n$ 貴金屬分割。特别地，第1貴金屬比例 ${\frac {1+{\sqrt {5}}}{2}}:1$ 称为黄金比例、第2貴金屬比例 $1+{\sqrt {2}}:1$ 称为白銀比例、第3貴金屬比例 ${\frac {3+{\sqrt {13}}}{2}}:1$ 称为青銅比例。 ^[1]

貴金属数

貴金属数
0	${\frac {0+{\sqrt {4}}}{2}}$	1	1
1	${\frac {1+{\sqrt {5}}}{2}}$	${\frac {1+{\sqrt {5}}}{2}}$	1.6180339887...
2	${\frac {2+{\sqrt {8}}}{2}}$	$1+{\sqrt {2}}$	2.4142135623...
3	${\frac {3+{\sqrt {13}}}{2}}$	${\frac {3+{\sqrt {13}}}{2}}$	3.3027756377...
4	${\frac {4+{\sqrt {20}}}{2}}$	$2+{\sqrt {5}}$	4.2360679774...
5	${\frac {5+{\sqrt {29}}}{2}}$	${\frac {5+{\sqrt {29}}}{2}}$	5.1925824035...
6	${\frac {6+{\sqrt {40}}}{2}}$	$3+{\sqrt {10}}$	6.1622776601...
7	${\frac {7+{\sqrt {53}}}{2}}$	${\frac {7+{\sqrt {53}}}{2}}$	7.1400549446...
8	${\frac {8+{\sqrt {68}}}{2}}$	$4+{\sqrt {17}}$	8.1231056256...
9	${\frac {9+{\sqrt {85}}}{2}}$	${\frac {9+{\sqrt {85}}}{2}}$	9.1097722286...
n	${\frac {n+{\sqrt {n^{2}+4}}}{2}}$

貴金属数是

{\frac {n+{\sqrt {n^{2}+4}}}{2}}

即二次方程式 $x^{2}-nx-1=0$ 的正根。

連分数

貴金属数的連分数表示是：

n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{\ddots }}}}}}}}=[n;n,n,n,n,\dots ]

数列的商的極限

黄金数（第1貴金属数）是斐波那契数列相邻两项的比的极限，白银数（第2貴金属数）是佩尔数列相邻两项的比的极限；一般地，也存在以第 $n$ 貴金属数为相邻两项的比的极限的数列。

数列 $\{M_{k}\}$ 的递推关系式

M_{0}=0,\quad M_{1}=1,\quad M_{k+2}=nM_{k+1}+M_{k}

一旦定义了此关系式，则在此之中，第 $n$ 貴金属数为 $\mu$ ，有

M_{k}={\frac {\mu ^{k}-(-\mu )^{-k}}{\mu +\mu ^{-1}}}={\frac {\mu ^{k}-(-\mu )^{-k}}{\sqrt {n^{2}+4}}}

成立。在这种情况下，这个序列的两个相邻项的商数在 $K\rightarrow \infty$ 收敛于 $\mu$ 。即

\lim _{k\to \infty }{\frac {M_{k+1}}{M_{k}}}=\mu

成立。

参考文献

^ # デザインの基礎、黄金比から大和比、第2黄金比まで. [2012年11月1日]. （原始内容存档于2021年2月27日）（日语）.

参见

查论编貴金屬比例
皮索數黄金比例角进制矩形菱形螺線（英语：Golden spiral）三角形黄金分割搜索斐波那契数开普勒三角白銀比例佩尔数青銅比例

查论编分數 & 比率
	除法＆比例	被除數 : 除數 = 商數
	分數	分子/分母=商數
	代數長寬比连分数十进制二进分数古埃及分數黄金分割率白銀比例整数最简分数精簡最小公分母音程百分比單位分數

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