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A156035 Decimal expansion of 3 + 2*sqrt(2). 127
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Lim_{n -> infinity} b(n+1)/b(n) = 3+2*sqrt(2) for b = A155464, A155465, A155466.

Lim_{n -> infinity} b(n)/b(n-1) = 3+2*sqrt(2) for b = A001652, A001653, A002315, A156156, A156157, A156158. - Klaus Brockhaus, Sep 23 2009

From Richard R. Forberg, Aug 14 2013: (Start)

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.

Ratios of 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).

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)

Let ABCD be a square inscribed in a circle. When P is the midpoint 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

Ratio between successive terms of A001652. - Harvey P. Dale, Jun 16 2017

LINKS

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Mathematical Reflections, Solution to Problem J286, Issue 1, 2014, p. 5.

FORMULA

Equals 1+A090488 = 3+A010466. - R. J. Mathar, Feb 19 2009

Equals exp(arccosh(3)), since arccosh(x) = log(x+sqrt(x^2-1)). - Stanislav Sykora, Nov 01 2013

Equals (1+sqrt(2))^2, that is, A014176^2. - Michel Marcus, May 08 2016

EXAMPLE

3 + 2*sqrt(2) = 5.828427124746190097603377448...

MATHEMATICA

RealDigits[N[3+2*Sqrt[2], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

PROG

(PARI) 3+sqrt(8) \\ Charles R Greathouse IV, Jun 10 2011

(MAGMA) SetDefaultRealField(RealField(100));  3 + 2*Sqrt(2); // G. C. Greubel, Aug 21 2018

CROSSREFS

Cf. A002193 (sqrt(2)), A090488, A010466, A014176.

Cf. A155464, A155465, A155466.

Cf. A104178 (decimal expansion of log_10(3+2*sqrt(2))).

Cf. A001109, A001541, A001542, A005319, A075870, A038723, A038725, A038761, A054488, A054489, A075848, A077413, A106328.

Sequence in context: A260061 A199379 A198844 * A284697 A319261 A010489

Adjacent sequences:  A156032 A156033 A156034 * A156036 A156037 A156038

KEYWORD

cons,easy,nonn

AUTHOR

Klaus Brockhaus, Feb 02 2009

STATUS

approved

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Last modified November 21 01:33 EST 2019. Contains 329349 sequences. (Running on oeis4.)