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%I #120 Mar 21 2024 17:45:18
%S 5,8,2,8,4,2,7,1,2,4,7,4,6,1,9,0,0,9,7,6,0,3,3,7,7,4,4,8,4,1,9,3,9,6,
%T 1,5,7,1,3,9,3,4,3,7,5,0,7,5,3,8,9,6,1,4,6,3,5,3,3,5,9,4,7,5,9,8,1,4,
%U 6,4,9,5,6,9,2,4,2,1,4,0,7,7,7,0,0,7,7,5,0,6,8,6,5,5,2,8,3,1,4,5,4,7,0,0,2
%N Decimal expansion of 3 + 2*sqrt(2).
%C Limit_{n -> oo} b(n+1)/b(n) = 3+2*sqrt(2) for b = A155464, A155465, A155466.
%C Limit_{n -> oo} b(n)/b(n-1) = 3+2*sqrt(2) for b = A001652, A001653, A002315, A156156, A156157, A156158. - _Klaus Brockhaus_, Sep 23 2009
%C From _Richard R. Forberg_, Aug 14 2013: (Start)
%C Ratios b(n+1)/b(n) for all sequences of the form b(n) = 6*b(n-1) - b(n-2), for any initial values of b(0) and b(1), converge to this ratio.
%C Ratios b(n+1)/b(n) for all sequences of the form b(n) = 5*b(n-1) + 5*b(n-2) + b(n-3), for all b(0), b(1) and b(2) also converge to 3 + 2*sqrt(2). For example see A084158 (Pell Triangles).
%C Ratios of alternating values, b(n+2)/b(n), for all sequences of the form b(n) = 2*b(n-1) + b(n-2), also converge to 3 + 2*sqrt(2). These include A000129 (Pell Numbers). Also see A014176. (End)
%C Let ABCD be a square inscribed in a circle. When P is the midpoint of the arc AB, then the ratio (PC*PD)/(PA*PB) is equal to 3+2*sqrt(2). See the Mathematical Reflections link. - _Michel Marcus_, Jan 10 2017
%C Limit of ratios of successive terms of A001652 when n-> infinity. - _Harvey P. Dale_, Jun 16 2017; improved by _Bernard Schott_, Feb 28 2022
%C A quadratic integer with minimal polynomial x^2 - 6x + 1. - _Charles R Greathouse IV_, Jul 11 2020
%C Ratio between radii of the large circumscribed circle R and the small internal circle r drawn on the Sangaku tablet at Isaniwa Jinjya shrine in Ehime Prefecture (pictures in links). - _Bernard Schott_, Feb 25 2022
%D Diogo Queiros-Condé and Michel Feidt, Fractal and Trans-scale Nature of Entropy, Iste Press and Elsevier, 2018, page 45.
%H Ivan Panchenko, <a href="/A156035/b156035.txt">Table of n, a(n) for n = 1..1000</a>
%H Mathematical Reflections, <a href="https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2014-01/mr_6_2013_solutions.pdf">Solution to Problem J286</a>, Issue 1, 2014, p. 5.
%H Bernard Schott, <a href="/A156035/a156035.png">Sangaku at Isaniwa Jinya</a>, The six circles.
%H Terakoya Suzu, <a href="https://terakoya-suzu.weebly.com/blog/sangaku-mathematics-tablet-ii">Sangaku (mathematics tablet) II</a>, Sangaku at Isaniwa Jinya shrine.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sangaku">Sangaku</a>.
%H Bernard Ycart, <a href="https://membres-ljk.imag.fr/Bernard.Ycart/mel/pe/node21.html">Les Sangakus</a>, Sangaku du Temple Isaniwa Jinya (in French).
%H <a href="/index/Al#algebraic_0">Index entries for algebraic numbers, degree </a>
%F Equals 1 + A090488 = 3 + A010466. - _R. J. Mathar_, Feb 19 2009
%F Equals exp(arccosh(3)), since arccosh(x) = log(x+sqrt(x^2-1)). - _Stanislav Sykora_, Nov 01 2013
%F Equals (1+sqrt(2))^2, that is, A014176^2. - _Michel Marcus_, May 08 2016
%F The periodic continued fraction is [5; [1, 4]]. - _Stefano Spezia_, Mar 17 2024
%e 3 + 2*sqrt(2) = 5.828427124746190097603377448...
%t RealDigits[N[3+2*Sqrt[2],200]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2011 *)
%o (PARI) 3+sqrt(8) \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Magma) SetDefaultRealField(RealField(100)); 3 + 2*Sqrt(2); // _G. C. Greubel_, Aug 21 2018
%Y Cf. A002193 (sqrt(2)), A090488, A010466, A014176.
%Y Cf. A155464, A155465, A155466.
%Y Cf. A104178 (decimal expansion of log_10(3+2*sqrt(2))).
%Y Cf. A000129, A001109, A001541, A001542, A001652, A001653, A002315, A005319, A075870, A038723, A038725, A038761, A054488, A054489, A075848, A077413, A084158, A106328, A156156, A156157, A156158.
%Y Cf. A242412 (sangaku).
%K cons,easy,nonn
%O 1,1
%A _Klaus Brockhaus_, Feb 02 2009
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