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A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio. 3
7, 8, 6, 1, 5, 1, 3, 7, 7, 7, 5, 7, 4, 2, 3, 2, 8, 6, 0, 6, 9, 5, 5, 8, 5, 8, 5, 8, 4, 2, 9, 5, 8, 9, 2, 9, 5, 2, 3, 1, 2, 2, 0, 5, 7, 8, 3, 7, 7, 2, 3, 2, 3, 7, 6, 6, 4, 9, 0, 1, 9, 7, 0, 1, 0, 1, 1, 8, 2, 0, 4, 7, 6, 2, 2, 3, 1, 0, 9, 1, 3, 7, 1, 1, 9, 1, 2, 8, 8, 9, 1, 5, 8, 5, 0, 8, 1, 3, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The hyperbolas y^2-x^2=1 and xy=1 meet at (1/c,c) and (-1/c,c), where c=sqrt(golden ratio); see the Mathematica program for a graph; see A189339 for hyperbolas meeting at (c,1/c) and (-c,-1/c).

Positive real root of x^4+x^2-1=0. - Paolo P. Lava, Oct 10 2013

This number is the eccentricity of an ellipse inscribed in a golden rectangle. - Jean-François Alcover, Sep 03 2015

c/sqrt(-1) is the limit of Pi(a;n)/2 := a^n * sqrt(a - f(a;n))  with f(a;0) = 0, and f(a;n) = sqrt(a + f(a;n-1)) for n >= 1, if one takes a = 1. For a=2 this gives Viète's formula for Pi/2 (see A019669). - Wolfdieter Lang, Jul 06 2018

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000

FORMULA

c = sqrt(1/phi) = sqrt(phi-1), with phi = A001622.

EXAMPLE

c=0.786151377757423286069558585842958929523122057...

MATHEMATICA

N[1/Sqrt[GoldenRatio], 110]

RealDigits[%]

FindRoot[x*Sqrt[1 + x^2] == 1, {x, 1.2, 1.3}, WorkingPrecision -> 110]

Plot[{Sqrt[1 + x^2], 1/x}, {x, 0, 3}]

PROG

(PARI) sqrt(2/(1+sqrt(5))) \\ Michel Marcus, Sep 03 2015

CROSSREFS

Cf. A001622, A019669, A139339.

Sequence in context: A262898 A256045 A004496 * A181624 A277683 A143300

Adjacent sequences:  A197759 A197760 A197761 * A197763 A197764 A197765

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 19 2011

STATUS

approved

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Last modified September 25 20:53 EDT 2018. Contains 315425 sequences. (Running on oeis4.)