Silver Ratio

The silver ratio is the quantity defined by the continued fraction

			(1)
			(2)

(Wall 1948, p. 24). It follows that

(3)

(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 , with the silver ratio being one exception.

The more general expressions

(5)

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

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More things to try:

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.