A358190 - OEIS

%I #10 Dec 11 2022 10:30:42

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

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

%U 3,0,8,9,1,5,1,0,5,4

%N Decimal expansion of the positive real root r of x^4 - 2*x - 1.

%C The two real and two complex roots are given by the negative roots of x^4 + 2*x - 1 (see A358189).

%F r = (sqrt(2)*u + sqrt(-2*u^2 + 2*sqrt(2*u)))/(2*sqrt(u)), where u = (((3/4)*( 9 + sqrt(129)))^(1/3) +  w1*((3/4)*( 9 - sqrt(129)))^(1/3))/3  = 0.4238537990..., and w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3), is one of the complex roots of x^3 - 1. Alternatively u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))).  A358189 uses the same u.

%e 1.3953369944670730187931436130710553428418349124097566207933090113521308...

%p a := ((3*sqrt(3) + sqrt(43))/4)^(1/3): b := (a - 1/a)/sqrt(3):

%p c := (sqrt(sqrt(2*b) - b^2) + b)/(sqrt(2*b)): evalf(c, 76); # _Peter Luschny_, Dec 10 2022

%t RealDigits[x /. FindRoot[x^4 - 2*x - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* _Amiram Eldar_, Dec 07 2022 *)

%Y Cf. A358189.

%K nonn,cons,easy

%O 1,2

%A _Wolfdieter Lang_, Dec 07 2022