A357472 - OEIS

%I #16 Nov 09 2022 05:13:09

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

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

%U 7,1,6,6,8,4,7,0,8

%N Decimal expansion of the real root of x^3 + x^2 + 2*x - 1.

%C This equals r0 - 1/3 where r0 is the real root of y^3 + (5/3)*y - 43/27.

%C The other roots of x^3 - x^2 + 2*x - 1 are (-1 + w1*((43 + 9*sqrt(29))/2)^(1/3) + ((43 - 9*sqrt(29))/2)^(1/3))/3 = -0.6963233908... + 1.4359498641...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 is one of the complex conjugate roots of x^3 - 1.

%C Using hyperbolic functions these roots are -(1 + sqrt(5)*(sinh((1/3)*arcsinh( (43/50)*sqrt(5))) - sqrt(3)*cosh((1/3)*arcsinh((43/50)*sqrt(5)))*i))/3, and its complex conjugate.

%F r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) - 20*(4*(43 + 9*sqrt(29)))^(-1/3))/6.

%F r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) + w1*(4*(43 - 9*sqrt(29)))^(1/3))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i) is one of the complex conjugate roots of x^3 - 1.

%F r = (-1 + 2*sqrt(5)*sinh((1/3)*arcsinh((43/50)*sqrt(5))))/3.

%F r = (-5*u^(1/3) + u^(-1/3) - 1)/3 where u = 2/(43 + 9*sqrt(29)). - _Peter Luschny_, Nov 01 2022

%F r = (-1/3) + (43/45) * Hyper2F1([1/3, 2/3], [3/2], -43^2/(5*10^2)). - _Gerry Martens_, Nov 04 2022

%e 0.3926467817026408117648795948843412507037649685934825897311396498445171668...

%p Digits:=100: u := 2/(43 + 9*sqrt(29)): (-5*u^(1/3) + u^(-1/3) - 1)/3:

%p evalf(%*10^78): ListTools:-Reverse(convert(floor(%), base, 10)); # _Peter Luschny_, Nov 01 2022

%t RealDigits[x /. FindRoot[x^3 + x^2 + 2*x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 26 2022 *)

%Y Cf. A160389, A255524, A255249, A357470, A357471.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Oct 25 2022