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A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio.

%I A197762 #50 Jul 16 2025 01:10:48
%S A197762 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,
%T A197762 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,
%U A197762 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
%N A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio.
%C A197762 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).
%C A197762 This number is the eccentricity of an ellipse inscribed in a golden rectangle. - _Jean-François Alcover_, Sep 03 2015
%C A197762 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
%H A197762 Chai Wah Wu, <a href="/A197762/b197762.txt">Table of n, a(n) for n = 0..10000</a>
%F A197762 Equals sqrt(1/phi) = sqrt(phi-1), with phi = A001622.
%F A197762 From _Amiram Eldar_, Feb 07 2022: (Start)
%F A197762 Equals 1/A139339.
%F A197762 Equals tan(arcsin(1/phi)).
%F A197762 Equals sin(arccos(1/phi)).
%F A197762 Equals cos(arcsin(1/phi)).
%F A197762 Equals cot(arccos(1/phi)). (End)
%e A197762 0.786151377757423286069558585842958929523122057...
%t A197762 N[1/Sqrt[GoldenRatio], 110]
%t A197762 RealDigits[%]
%t A197762 FindRoot[x*Sqrt[1 + x^2] == 1, {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t A197762 Plot[{Sqrt[1 + x^2], 1/x}, {x, 0, 3}]
%o A197762 (PARI) sqrt(2/(1+sqrt(5))) \\ _Michel Marcus_, Sep 03 2015
%o A197762 (PARI) my(c=1/quadgen(5)); a_vector(len) = digits(sqrtint(floor(c*100^len))); \\ _Kevin Ryde_, Jul 12 2025
%Y A197762 Cf. A001622, A019669, A139339.
%K A197762 nonn,cons
%O A197762 0,1
%A A197762 _Clark Kimberling_, Oct 19 2011