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A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio. 5
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).
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
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
Sequence in context: A198365 A262898 A004496 * A181624 A277683 A143300
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 19 2011
STATUS
approved

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Last modified May 18 02:04 EDT 2024. Contains 372615 sequences. (Running on oeis4.)