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+\\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 $P_{n}\\delta_{S}+P_{n-1}$ .

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