cp's OEIS Frontend

A357472 Decimal expansion of the real root of x^3 + x^2 + 2*x - 1.

%I A357472 #16 Nov 09 2022 05:13:09
%S A357472 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 A357472 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 A357472 7,1,6,6,8,4,7,0,8
%N A357472 Decimal expansion of the real root of x^3 + x^2 + 2*x - 1.
%C A357472 This equals r0 - 1/3 where r0 is the real root of y^3 + (5/3)*y - 43/27.
%C A357472 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 A357472 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 A357472 r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) - 20*(4*(43 + 9*sqrt(29)))^(-1/3))/6.
%F A357472 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 A357472 r = (-1 + 2*sqrt(5)*sinh((1/3)*arcsinh((43/50)*sqrt(5))))/3.
%F A357472 r = (-5*u^(1/3) + u^(-1/3) - 1)/3 where u = 2/(43 + 9*sqrt(29)). - _Peter Luschny_, Nov 01 2022
%F A357472 r = (-1/3) + (43/45) * Hyper2F1([1/3, 2/3], [3/2], -43^2/(5*10^2)). - _Gerry Martens_, Nov 04 2022
%e A357472 0.3926467817026408117648795948843412507037649685934825897311396498445171668...
%p A357472 Digits:=100: u := 2/(43 + 9*sqrt(29)): (-5*u^(1/3) + u^(-1/3) - 1)/3:
%p A357472 evalf(%*10^78): ListTools:-Reverse(convert(floor(%), base, 10)); # _Peter Luschny_, Nov 01 2022
%t A357472 RealDigits[x /. FindRoot[x^3 + x^2 + 2*x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 26 2022 *)
%Y A357472 Cf. A160389, A255524, A255249, A357470, A357471.
%K A357472 nonn,cons,easy
%O A357472 0,1
%A A357472 _Wolfdieter Lang_, Oct 25 2022