A197762 - OEIS

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

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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...