A337569 - OEIS

%I #47 Dec 21 2024 22:32:03

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

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

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

%N Decimal expansion of the real solution to x^3 = 3 - x.

%C x = (3 - (3 - (3 - ...)^(1/3))^(1/3))^(1/3).

%C The other two solutions are (w1*(81/2 + (3/2)*sqrt(741))^(1/3) + (81/2 - (3/2)*sqrt(741))^(1/3))/3 = -0.60670583... + 1.45061225...*i, where w1 = (-1 + sqrt(3)*i)/2, and its complex conjugate. With hyperbolic functions these solutions are -(1/3)*sqrt(3)*(sinh((1/3)*arcsinh((9/2)*sqrt(3))) - sqrt(3)*cosh((1/3)*arcsinh((9/2)*sqrt(3)))*i), and its complex conjugate. - _Wolfdieter Lang_, Sep 13 2022

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

%F Equals (3/2 + sqrt(741/324))^(1/3) - (-3/2 + sqrt(741/324))^(1/3).

%F From _Wolfdieter Lang_, Sep 13 2022: (Start)

%F Equals (1/6)*(324 + 12*sqrt(741))^(1/3) - 2/(324 + 12*sqrt(741))^(1/3).

%F Equals ((81/2 + (3/2)*sqrt(741))^(1/3) + w1*(81/2 - (3/2)*sqrt(741))^(1/3))/3, with w1 = (-1 + sqrt(3)*i)/2, one of the complex roots of x^3 - 1.

%F Equals (2/3)*sqrt(3)*sinh((1/3)*arcsinh((9/2)*sqrt(3))). (End)

%e 1.2134116627622296...

%p Digits:=100; solve(x^3+x-3=0); evalf(%)[1];

%t RealDigits[x /. FindRoot[x^3 + x - 3, {x, 1}, WorkingPrecision -> 100], 10, 90][[1]] (* _Amiram Eldar_, Sep 03 2020 *)

%o (PARI) solve(n=0,2,n^3+n-3)

%o (PARI) polroots(n^3+n-3)[1]

%o (PARI) polrootsreal(x^3+x-3)[1] \\ _Charles R Greathouse IV_, Oct 27 2023

%o (MATLAB) format long; solve('x^3+x-3=0'); ans(1), (eval(ans))

%Y Cf. A294644, A357102.

%K nonn,cons

%O 1,2

%A _Michal Paulovic_, Sep 01 2020