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

Author

Clark Kimberling, Oct 19 2011

Keywords

• nonn
• cons

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

Examples

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Links

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

Crossrefs

Cf. A001622, A019669, A139339.

Programs

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}]

PARI

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

PARI

my(c=1/quadgen(5)); a_vector(len) = digits(sqrtint(floor(c*100^len))); \\ Kevin Ryde, Jul 12 2025

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)