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%I #122 Mar 25 2024 21:39:51
%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
%C This algebraic integer of degree 2, with minimal polynomial x^2 - 2*x - 1, is also the length ratio diagonal/side of the second largest diagonal in the regular octagon (not counting the side). The other two diagonal/side ratios are A179260 and A121601. - _Wolfdieter Lang_, Oct 28 2020
%C c^n = A001333(n) + A000129(n) * sqrt(2). - _Gary W. Adamson_, Apr 26 2023
%C c^n = c * A000129(n) + A000129(n-1), where c = 1 + sqrt(2). - _Gary W. Adamson_, Aug 30 2023
%D B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
%H G. C. Greubel, <a href="/A014176/b014176.txt">Table of n, a(n) for n = 1..10000</a>
%H Nicholas R. Beaton, Mireille Bousquet-Mélou, Jan de Gier, Hugo Duminil-Copin, and Anthony J. Guttmann, <a href="https://arxiv.org/abs/1109.0358">The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1+sqrt(2)</a>, arXiv:1109.0358 [math-ph], 2011-2013.
%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="https://en.wikipedia.org/wiki/silver_ratio">Silver ratio</a>
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F Conjecture: 1+sqrt(2) = lim_{n->oo} A179807(n+1)/A179807(n).
%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 Limit_{n->oo} exp(asinh(cos(Pi/n))) = sqrt(2) + 1. - _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
%F Equals Product_{k>=1} A047621(k) / A047522(k) = (3/1) * (5/7) * (11/9) * (13/15) * (19/17) * (21/23) * ... . - _Dimitris Valianatos_, Mar 27 2019
%F From _Wolfdieter Lang_, Nov 10 2023:(Start)
%F Equals lim_{n->oo} A000129(n+1)/A000129(n) (see A000129, Pell).
%F Equals lim_{n->oo} S(n+1, 2*sqrt(2))/S(n, 2*sqrt(2)), with the Chebyshev S(n,x) polynomial (see A049310). (End)
%F From _Peter Bala_, Mar 24 2024: (Start)
%F An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 8*k + 6 for k >= 0.
%F For example, taking k = 0 and k = 1 yields
%F sqrt(2) + 1 = 15/(6 + (1*3)/(12 + (5*7)/(12 + (9*11)/(12 + (13*15)/(12 + ... + (4*n + 1)*(4*n + 3)/(12 + ... )))))) and
%F sqrt(2) + 1 = (715/21) * 1/(14 + (1*3)/(28 + (5*7)/(28 + (9*11)/(28 + (13*15)/(28 + ... + (4*n + 1)*(4*n + 3)/(28 + ... )))))). (End)
%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 Apart from initial digit the same as A002193.
%Y Cf. A000032, A006497, A080039, A179260, A121601.
%Y See A098316 for [3;3,3,...]; A098317 for [4;4,4,...]; A098318 for [5;5,5,...]. - _Hieronymus Fischer_, Oct 20 2010
%Y Cf. A000129, A049310.
%K nonn,cons,easy
%O 1,1
%A _N. J. A. Sloane_
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