A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio.

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

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).

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

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

From Amiram Eldar, Feb 07 2022: (Start)

Equals 1/A139339.

Equals tan(arcsin(1/phi)).

Equals sin(arccos(1/phi)).

Equals cos(arcsin(1/phi)).

Equals cot(arccos(1/phi)). (End)

Example

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: A198365 A262898 A004496 * A181624 A277683 A143300

Adjacent sequences: A197759 A197760 A197761 * A197763 A197764 A197765

Keyword

nonn,cons

Author

Clark Kimberling, Oct 19 2011

Status

approved