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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 ${frac(x^n)}_(n=1)^infty$ of power fractional parts, where $frac(x)$ is the fractional part, is equidistributed for real numbers outside a set of Lebesgue measure zero (Koksma 1935, p. 250). The silver ratio is one exception, since its powers approach integers.

For a positive integer , 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." https://r-knott.surrey.ac.uk/Fibonacci/cfINTRO.html#silver.Koksma, J. F. "Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins." Compositio Math. 2, 250-258, 1935. https://www.numdam.org/item/CM_1935__2__250_0/.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.