# 貴金屬比例

${\displaystyle 1:{\frac {n+{\sqrt {n^{2}+4}}}{2}}}$（n为自然数）

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

## 貴金属数

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

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

### 連分数

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

### 数列的商的極限

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

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

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

## 参考文献

1. ^ # 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）.