Metallic mean

The metallic mean (also metallic ratio, metallic constant, or noble mean) of a natural number n is a positive real number, denoted here S n , {\displaystyle S_{n},} that satisfies the following equivalent characterizations: the unique positive real number x {\displaystyle x} such that x = n + 1 x {\textstyle x=n+{\frac {1}{x}}} the positive root of the quadratic equation x 2 − n x − 1 = 0 {\displaystyle x^{2}-nx-1=0} the number n + n 2 + 4 2 = 2 n 2 + 4 − n {\textstyle {\frac {n+{\sqrt {n^{2}+4}}}{2}}={\frac {2}{{\sqrt {n^{2}+4}}-n}}} the number whose expression as a continued fraction is [ n ; n , n , n , n , … ] = n + 1 n + 1 n + 1 n + 1 n + ⋱ {\displaystyle [n;n,n,n,n,\dots ]=n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{n+\ddots \,}}}}}}}}} Metallic means are (successive) derivations of the golden ( n = 1 {\displaystyle n=1} ) and silver ratios ( n = 2 {\displaystyle n=2} ), and share some of their interesting properties. The term "bronze ratio" ( n = 3 {\displaystyle n=3} ) (cf. Golden Age and Olympic Medals) and even metals such as copper ( n = 4 {\displaystyle n=4} ) and nickel ( n = 5 {\displaystyle n=5} ) are occasionally found in the literature. In terms of algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than 1 {\displaystyle 1} and have − 1 {\displaystyle -1} as their norm. The defining equation x 2 − n x − 1 = 0 {\displaystyle x^{2}-nx-1=0} of the nth metallic mean is the characteristic equation of a linear recurrence relation of the form x k = n x k − 1 + x k − 2 . {\displaystyle x_{k}=nx_{k-1}+x_{k-2}.} It follows that, given such...