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A014176
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Decimal expansion of the silver mean, 1+sqrt(2).
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21
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2, 4, 1, 4, 2, 1, 3, 5, 6, 2, 3, 7, 3, 0, 9, 5, 0, 4, 8, 8, 0, 1, 6, 8, 8, 7, 2, 4, 2, 0, 9, 6, 9, 8, 0, 7, 8, 5, 6, 9, 6, 7, 1, 8, 7, 5, 3, 7, 6, 9, 4, 8, 0, 7, 3, 1, 7, 6, 6, 7, 9, 7, 3, 7, 9, 9, 0, 7, 3, 2, 4, 7, 8, 4, 6, 2, 1, 0, 7, 0, 3, 8, 8, 5, 0, 3, 8, 7, 5, 3, 4, 3, 2, 7, 6, 4, 1, 5, 7
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| In terms of continued fractions the constant c can be described by c=[2;2,2,2,...]. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), October 20 2010
Side length of smallest square containing five circles of diameter 1. [Charles R Greathouse IV, Apr 05, 2011]
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LINKS
| Anonymous, Exact trigonometric constants, Wikipedia
Eric Weisstein's World of Mathematics, Silver Ratio
Wikipedia, Silver ratio
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FORMULA
| Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 02 2009 (Start): Set c:=1+sqrt(2). Then the fractional part of c^n equals 1/c^n, if n odd. For even n, the fractional part of c^n is equal to 1-(1/c^n).
c:=1+sqrt(2) satisfies c-c^(-1)=floor(c)=2, hence c^n+(-c)^(-n)=nint(c^n) for n>0, which follows from the general formula of A001622.
1/c=sqrt(2)-1.
See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which suffice x-x^(-1)=floor(x).
Other examples of constants x satisfying the relation x-x^(-1)=floor(x) include A001622 (the golden ratio: where floor(x)=1) and A098316 (the "bronze" ratio: where floor(x)=3). (End)
1+sqrt(2)=A179807(n+1)/A179807(n) as n --> infinity. (conjecture)
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MATHEMATICA
| RealDigits[1 + Sqrt@ 2, 10, 111] (* Or *)
RealDigits[Exp@ ArcSinh@ 1, 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)
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CROSSREFS
| Cf. A002193, A000032, A006497, A080039.
See A098316 for [3;3,3,...]; A098317 for [4;4,4,...]; A098318 for [5;5,5,...]. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), October 20 2010
Sequence in context: A194733 A143973 A011167 * A060047 A135185 A201774
Adjacent sequences: A014173 A014174 A014175 * A014177 A014178 A014179
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KEYWORD
| nonn,cons,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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