A088559 Decimal expansion of R^2 where R^2 is the real root of x^3 + 2*x^2 + x - 1 = 0.

4, 6, 5, 5, 7, 1, 2, 3, 1, 8, 7, 6, 7, 6, 8, 0, 2, 6, 6, 5, 6, 7, 3, 1, 2, 2, 5, 2, 1, 9, 9, 3, 9, 1, 0, 8, 0, 2, 5, 5, 7, 7, 5, 6, 8, 4, 7, 2, 2, 8, 5, 7, 0, 1, 6, 4, 3, 1, 8, 3, 1, 1, 1, 2, 4, 9, 2, 6, 2, 9, 9, 6, 6, 8, 5, 0, 1, 7, 8, 4, 0, 4, 7, 8, 1, 2, 5, 8, 0, 1, 1, 9, 4, 9, 0, 9, 2, 7, 0, 0, 6, 4, 3, 8

Offset: 0

Author

Benoit Cloitre, Nov 19 2003

Keywords

• cons
• nonn

Comments

Arise in a study of AGM (arithmetic-geometric mean) and HGM (harmonic-geometric mean) - like sequences. Let u(k+1)=sqrt(u(k)*v(k)); v(k+1)=v(k)+u(k) and r(k+1)=sqrt(r(k)*s(k)); s(k+1)=1/(1/r(k)+1/s(k)). Then for any positive initial values u(0),v(0),r(0),s(0) limit k-->oo u(k)/v(k)= limit k-->oo s(k)/r(k)=R^2.

From Wolfdieter Lang, Nov 07 2022: (Start)

This equals r0 - 2/3 where r0 is the real root of y^3 - (1/3)*y - 29/27.

The other roots of x^3 + 2*x^2 + x - 1 are (-2 + w1*((29 + 3*sqrt(93))/2)^(1/3) + w2*((29 - 3*sqrt(93))/2)^(1/3))/3 = -1.2327856159... + 0.7925519925...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex conjugate roots of x^3 - 1.

Using hyperbolic functions these roots are (-2 - cosh((1/3)*arccosh(29/2)) + sqrt(3)*sinh((1/3)*arccosh(29/2))*i)/3, and its complex conjugate. (End)

Examples

			0.465571231876768026656731225219939108025577568472285701643183111249262996685...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A092526, A144229, A358181.

Programs

Mathematica

Root[x^3 + 2x^2 + x - 1, 1] // RealDigits[#, 10, 104]& // First (* Jean-François Alcover, Mar 04 2013 *)

PARI

allocatemem(932245000); default(realprecision, 20080); x=10*solve(x=0, 1, x^3 + 2*x^2 + x - 1); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b088559.txt", n, " ", d)); \\ Harry J. Smith, Jun 21 2009

PARI

polrootsreal(x^3 + 2*x^2 + x - 1)[1] \\ Charles R Greathouse IV, Mar 03 2016

Formula

R^2=0.46557123187676... 1+R^2=1.46557123187676... = A092526 constant.

From Vaclav Kotesovec, Dec 18 2014: (Start)

Equals (1/6)*(116+12*sqrt(93))^(1/3) + 2/(3*(116+12*sqrt(93))^(1/3)) - 2/3.

Equals 2*cos(arccos(29/2)/3)/3 - 2/3.

Equals A092526 - 1.

(End)

From Wolfdieter Lang, Nov 07 2022: (Start)

Equals (-2 + ((29 + 3*sqrt(93))/2)^(1/3) + ((29 + 3*sqrt(93))/2)^(-1/3))/3.

Equals (-2 + ((29 + 3*sqrt(93))/2)^(1/3) + ((29 - 3*sqrt(93))/2)^(1/3))/3.

Also with hperbolic cosh and arccosh instead of cos and arccos above.

(End)