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A139339 Decimal expansion of the square root of the golden ratio. 30
1, 2, 7, 2, 0, 1, 9, 6, 4, 9, 5, 1, 4, 0, 6, 8, 9, 6, 4, 2, 5, 2, 4, 2, 2, 4, 6, 1, 7, 3, 7, 4, 9, 1, 4, 9, 1, 7, 1, 5, 6, 0, 8, 0, 4, 1, 8, 4, 0, 0, 9, 6, 2, 4, 8, 6, 1, 6, 6, 4, 0, 3, 8, 2, 5, 3, 9, 2, 9, 7, 5, 7, 5, 5, 3, 6, 0, 6, 8, 0, 1, 1, 8, 3, 0, 3, 8, 4, 2, 1, 4, 9, 8, 8, 4, 6, 0, 2, 5, 8, 5, 3, 8, 5, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The hyperbolas x^2 - y^2 = 1 and xy = 1 meet at (c, 1/c) and (-c, -1/c), where c = sqrt(golden ratio); see the Mathematica program for a graph. - Clark Kimberling, Oct 19 2011
An algebraic integer of degree 4. Minimal polynomial: x^4 - x^2 - 1. - Charles R Greathouse IV, Jan 07 2013
Also the limiting value of the ratio of the slopes of the tangents drawn to the function y=sqrt(x) from the abscissa F(n) points (where F(n)=A000045(n) are the Fibonacci numbers and n > 0). - Burak Muslu, Apr 04 2021
The length of the base of the isosceles triangle of smallest perimeter which circumscribes a unit-diameter semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022
The unique real solution to arcsec(x) = arccot(x). - Wolfe Padawer, Apr 14 2023
REFERENCES
B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 45-48.
LINKS
Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176. Solution published in Vol. 12, No. 1 (Winter 2000), pp. 61-62.
Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
FORMULA
Equals sqrt((1 + sqrt(5))/2).
Equals 1/sqrt(A094214). - Burak Muslu, Apr 04 2021
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A197762.
Equals tan(arccos(1/phi)).
Equals cot(arcsin(1/phi)). (End)
From Gerry Martens, Jul 30 2023: (Start)
Equals 5^(1/4)*cos(arctan(2)/2).
Equals Re(sqrt(1+2*i)) (the imaginary part is A197762). (End)
EXAMPLE
1.2720196495140689642524224617374914917156080418400...
MAPLE
Digits:=100: evalf(sqrt((1+sqrt(5))/2)); # Muniru A Asiru, Sep 11 2018
MATHEMATICA
N[Sqrt[GoldenRatio], 100]
FindRoot[x*Sqrt[-1 + x^2] == 1, {x, 1.2, 1.3}, WorkingPrecision -> 110]
Plot[{Sqrt[-1 + x^2], 1/x}, {x, 0, 3}] (* Clark Kimberling, Oct 19 2011 *)
PROG
(PARI) sqrt((1+sqrt(5))/2) \\ Charles R Greathouse IV, Jan 07 2013
CROSSREFS
Cf. A000045, A001622, A094214, A104457, A098317, A002390; A197762 (related intersection of hyperbolas).
Sequence in context: A242207 A060465 A219177 * A090986 A245221 A195726
KEYWORD
nonn,cons,easy
AUTHOR
Mohammad K. Azarian, Apr 14 2008
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)