silver ratio

The silver ratio is the sum of 1 and the square root of 2, represented by the Greek letter delta with a subscript S. That is, ${\delta }_S=1+\sqrt{2}$, with an approximate value of 2.4142135623730950488 (see A014176 in Sloane’s OEIS). Its continued fraction is

$$2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\mathrm{\ddots }}}},$$

which suggests that the Pell numbers $P_n$ can be used as convergents. Similarly, the $n$th power of the silver ratio for $n>0$ is $P_n{\delta }_S+P_{n-1}$.

Title	silver ratio
Canonical name	SilverRatio
Date of creation	2013-03-22 16:42:10
Last modified on	2013-03-22 16:42:10
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 40A05