Silver Ratio
المؤلف:
Wall, H. S.
المصدر:
Analytic Theory of Continued Fractions. New York: Chelsea, 1948.
الجزء والصفحة:
...
23-1-2020
1181
Silver Ratio
The silver ratio is the quantity defined by the continued fraction
(Wall 1948, p. 24). It follows that
 |
(3)
|
so
 |
(4)
|
(OEIS A014176).
The sequence ![<span style=]()
{frac(x^n)}" src="http://mathworld.wolfram.com/images/equations/SilverRatio/Inline7.gif" style="height:15px; width:56px" />, of power fractional parts, where 
is the fractional part, is equidistributed for almost all real numbers 
, with the silver ratio being one exception.
The more general expressions
![[n,n,...]=1/2(n+sqrt(n^2+4))](http://mathworld.wolfram.com/images/equations/SilverRatio/NumberedEquation3.gif) |
(5)
|
are sometimes known in general as silver means (Knott). The first few values are summarized in the table below.
 |
OEIS |
![[n^_]](http://mathworld.wolfram.com/images/equations/SilverRatio/Inline11.gif) |
value |
| 1 |
A001622 |
 |
1.618033988... |
| 2 |
A014176 |
 |
2.414213562... |
| 3 |
A098316 |
 |
3.302775637... |
| 4 |
A098317 |
 |
4.236067977... |
| 5 |
A098318 |
 |
5.192582403... |
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.
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
