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A014176 Decimal expansion of the silver mean, 1+sqrt(2). 41

%I

%S 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,

%T 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,

%U 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

%N Decimal expansion of the silver mean, 1+sqrt(2).

%C From _Hieronymus Fischer_, Jan 02 2009: (Start)

%C 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 c:=1+sqrt(2) satisfies c-c^(-1)=floor(c)=2, hence c^n + (-c)^(-n) = round(c^n) for n>0, which follows from the general formula of A001622.

%C 1/c = sqrt(2)-1.

%C See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).

%C 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)

%C In terms of continued fractions the constant c can be described by c=[2;2,2,2,...]. - _Hieronymus Fischer_, Oct 20 2010

%C Side length of smallest square containing five circles of diameter 1. - _Charles R Greathouse IV_, Apr 05, 2011

%C Largest radius of four circles tangent to a circle of radius 1. - _Charles R Greathouse IV_, Jan 14 2013

%C An analog of Fermat theorem: for prime p, round(c^p) == 2 (mod p). - _Vladimir Shevelev_, Mar 02 2013

%C n*(1+sqrt(2)) is the perimeter of a 45-45-90 triangle with hypotenuse n. - _Wesley Ivan Hurt_, Apr 09 2016

%H G. C. Greubel, <a href="/A014176/b014176.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SilverRatio.html">Silver Ratio</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Exact_trigonometric_constants">Exact trigonometric constants</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Metallic_mean">Metallic mean</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/silver_ratio">Silver ratio</a>

%F 1+sqrt(2) = lim A179807(n+1)/A179807(n) as n -> infinity. (conjecture)

%F Equals cot(Pi/8) = tan(Pi*3/8). - _Bruno Berselli_, Dec 13 2012, and _M. F. Hasler_, Jul 08 2016

%F Silver mean = 2 + Sum_{n>=0} (-1)^n/(P(n-1)*P(n)), where P(n) is the n-th Pell number (A000129). - _Vladimir Shevelev_, Feb 22 2013

%F Equals exp(arcsinh(1)) which is exp(A091648). - _Stanislav Sykora_, Nov 01 2013

%F exp(asinh(cos(Pi/n))) -> sqrt(2)+1 as n -> infinity. - _Geoffrey Caveney_, Apr 23 2014

%F exp(asinh(cos(Pi/2 - log(sqrt(2)+1)*i)))= exp(asinh(sin(log(sqrt(2)+1)*i))) = i. - _Geoffrey Caveney_, Apr 23 2014

%e 2.414213562373095...

%p Digits:=100: evalf(1+sqrt(2)); # _Wesley Ivan Hurt_, Apr 09 2016

%t RealDigits[1 + Sqrt@ 2, 10, 111] (* Or *)

%t RealDigits[Exp@ ArcSinh@ 1, 10, 111][[1]] (* _Robert G. Wilson v_, Aug 17 2011 *)

%t Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[

%t Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}], {Blue, Circle[{0, 0}, 1]}]]] Circs[4] (* _Charles R Greathouse IV_, Jan 14 2013 *)

%o (PARI) 1+sqrt(2) \\ _Charles R Greathouse IV_, Jan 14 2013

%Y Cf. A002193, A000032, A006497, A080039.

%Y See A098316 for [3;3,3,...]; A098317 for [4;4,4,...]; A098318 for [5;5,5,...]. - _Hieronymus Fischer_, Oct 20 2010

%K nonn,cons,easy

%O 1,1

%A _N. J. A. Sloane_

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