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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 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

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