A139339 Decimal expansion of the square root of the golden ratio.

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

Offset: 1

Author

Mohammad K. Azarian, Apr 14 2008

Keywords

• nonn
• cons
• easy

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

Examples

			1.2720196495140689642524224617374914917156080418400...
		

References

• B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 45-48.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• 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.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A000045, A001622, A094214, A104457, A098317, A002390; A197762 (related intersection of hyperbolas).

Programs

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 *)

PARI

sqrt((1+sqrt(5))/2) \\ Charles R Greathouse IV, Jan 07 2013

PARI

a(n) = sqrtint(10^(2*n-2)*quadgen(5))%10; \\ Chittaranjan Pardeshi, Aug 24 2024

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)