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Silver Ratio


The silver ratio is the quantity defined by the continued fraction

delta_S=[2,2,2,...]
(1)
=2+1/(2+1/(2+1/(2+...)))
(2)

(Wall 1948, p. 24). It follows that

 (delta_S-1)^2=2,
(3)

so

 delta_S=sqrt(2)+1=2.41421...
(4)

(OEIS A014176).

The sequence {frac(x^n)}, of power fractional parts, where frac(x) is the fractional part, is equidistributed for almost all real numbers x>1, with the silver ratio being one exception.

The more general expressions

 [n,n,...]=1/2(n+sqrt(n^2+4))
(5)

are sometimes known in general as silver means (Knott). The first few values are summarized in the table below.

nOEIS[n^_]value
1A0016221/2(1+sqrt(5))1.618033988...
2A0141761+sqrt(2)2.414213562...
3A0983161/2(3+sqrt(13))3.302775637...
4A0983172+sqrt(5)4.236067977...
5A0983181/2(5+sqrt(29))5.192582403...

See also

Equidistributed Sequence, Golden Ratio, Golden Ratio Conjugate, Power Fractional Parts

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References

Knott, R. "The Silver Means." http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#silver.Sloane, N. J. A. Sequences A001622/M4046, A014176, A098316, A098317, and A098318 in "The On-Line Encyclopedia of Integer Sequences."Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.

Referenced on Wolfram|Alpha

Silver Ratio

Cite this as:

Weisstein, Eric W. "Silver Ratio." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SilverRatio.html

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