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A014176 Decimal expansion of the silver mean, 1+sqrt(2). 23
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; text; internal format)
OFFSET

1,1

COMMENTS

From Hieronymus Fischer, 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) = round(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 satisfy 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)

In terms of continued fractions the constant c can be described by c=[2;2,2,2,...]. - Hieronymus Fischer, Oct 20 2010

Side length of smallest square containing five circles of diameter 1. - Charles R Greathouse IV, Apr 05, 2011

Largest radius of four circles tangent to a circle of radius 1. - Charles R Greathouse IV, Jan 14 2013

An analog of Fermat theorem: for prime p, round(c^p) == 2 (mod p). - Vladimir Shevelev, Mar 02 2013

LINKS

Table of n, a(n) for n=1..99.

Eric Weisstein's World of Mathematics, Silver Ratio

Wikipedia, Exact trigonometric constants

Wikipedia, Silver ratio

FORMULA

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

Equals cot(Pi/8). [Bruno Berselli, Dec 13 2012]

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

Equals exp(arcsinh(1)) which is exp(A091648). - Stanislav Sykora, Nov 01 2013

exp(asinh(cos(Pi/n))) -> sqrt(2)+1 as n -> infinity. - Geoffrey Caveney, Apr 23 2014

exp(asinh(cos(Pi/2 - log(sqrt(2)+1)*i)))= exp(asinh(sin(log(sqrt(2)+1)*i))) = i. - Geoffrey Caveney, Apr 23 2014

MATHEMATICA

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

RealDigits[Exp@ ArcSinh@ 1, 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

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

  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 *)

PROG

(PARI) 1+sqrt(2) \\ Charles R Greathouse IV, Jan 14 2013

CROSSREFS

Cf. A002193, A000032, A006497, A080039.

See A098316 for [3;3,3,...]; A098317 for [4;4,4,...]; A098318 for [5;5,5,...]. - Hieronymus Fischer, Oct 20 2010

Sequence in context: A194733 A143973 A011167 * A060047 A135185 A201774

Adjacent sequences:  A014173 A014174 A014175 * A014177 A014178 A014179

KEYWORD

nonn,cons,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 28 20:04 EST 2014. Contains 250399 sequences.