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Decimal expansion of R^2 where R^2 is the real root of x^3 + 2*x^2 + x - 1 = 0.
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%I #41 Oct 27 2023 10:25:11

%S 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,

%T 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,

%U 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

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

%C 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.

%C From _Wolfdieter Lang_, Nov 07 2022: (Start)

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

%C 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.

%C 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)

%H Harry J. Smith, <a href="/A088559/b088559.txt">Table of n, a(n) for n = 0..20000</a>

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

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

%F From _Vaclav Kotesovec_, Dec 18 2014: (Start)

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

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

%F Equals A092526 - 1.

%F (End)

%F From _Wolfdieter Lang_, Nov 07 2022: (Start)

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

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

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

%F (End)

%e 0.465571231876768026656731225219939108025577568472285701643183111249262996685...

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

%o (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

%o (PARI) polrootsreal(x^3 + 2*x^2 + x - 1)[1] \\ _Charles R Greathouse IV_, Mar 03 2016

%Y Cf. A092526, A144229, A358181.

%K cons,nonn

%O 0,1

%A _Benoit Cloitre_, Nov 19 2003