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%I #69 Aug 16 2023 16:30:57
%S 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,
%T 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,
%U 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
%N Decimal expansion of the square root of the golden ratio.
%C 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
%C An algebraic integer of degree 4. Minimal polynomial: x^4 - x^2 - 1. - _Charles R Greathouse IV_, Jan 07 2013
%C 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
%C 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
%C The unique real solution to arcsec(x) = arccot(x). - _Wolfe Padawer_, Apr 14 2023
%D B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 45-48.
%H Chai Wah Wu, <a href="/A139339/b139339.txt">Table of n, a(n) for n = 1..10000</a>
%H Mohammad K. Azarian, <a href="https://doi.org/10.35834/1998/1003176">Problem 123</a>, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176. <a href="https://doi.org/10.35834/2000/1201050">Solution</a> published in Vol. 12, No. 1 (Winter 2000), pp. 61-62.
%H Duane W. DeTemple, <a href="https://www.fq.math.ca/Scanned/30-3/detemple.pdf">The Triangle of Smallest Perimeter which Circumscribes a Semicircle</a>, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F Equals sqrt((1 + sqrt(5))/2).
%F Equals 1/sqrt(A094214). - _Burak Muslu_, Apr 04 2021
%F From _Amiram Eldar_, Feb 07 2022: (Start)
%F Equals 1/A197762.
%F Equals tan(arccos(1/phi)).
%F Equals cot(arcsin(1/phi)). (End)
%F From _Gerry Martens_, Jul 30 2023: (Start)
%F Equals 5^(1/4)*cos(arctan(2)/2).
%F Equals Re(sqrt(1+2*i)) (the imaginary part is A197762). (End)
%e 1.2720196495140689642524224617374914917156080418400...
%p Digits:=100: evalf(sqrt((1+sqrt(5))/2)); # _Muniru A Asiru_, Sep 11 2018
%t N[Sqrt[GoldenRatio], 100]
%t FindRoot[x*Sqrt[-1 + x^2] == 1, {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t Plot[{Sqrt[-1 + x^2], 1/x}, {x, 0, 3}] (* _Clark Kimberling_, Oct 19 2011 *)
%o (PARI) sqrt((1+sqrt(5))/2) \\ _Charles R Greathouse IV_, Jan 07 2013
%Y Cf. A000045, A001622, A094214, A104457, A098317, A002390; A197762 (related intersection of hyperbolas).
%K nonn,cons,easy
%O 1,2
%A _Mohammad K. Azarian_, Apr 14 2008
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