A357100 - OEIS

%I #18 Nov 09 2022 04:40:24

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

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

%U 6,4,7,6,4,7,9,2

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

%C This equals r0 - 1/3 where r0 is the real root of y^3 - (1/3)*y - 79/27.

%C The other two roots are (w1*(79/2 + (9/2)*sqrt(77))^(1/3) + w2*(79/2 - (9/2)*sqrt(77))^(1/3) - 1)/3 = -1.08727970... + 1.1713121...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

%C Using hyperbolic function these roots are (-(1 + cosh((1/3)*arccosh(79/2))) + sqrt(3)*sinh((1/3)*arccosh(79/2))*i)/3, and its complex conjugate.

%F r = ((316 + 36*sqrt(77))^(1/3) + 4/(316 + 36*sqrt(77))^(1/3) - 2)/6.

%F r = ((79/2 + (9/2)*sqrt(77))^(1/3) + (79/2 - (9/2)*sqrt(77))^(1/3) - 1)/3.

%F r = (2*cosh((1/3)*arccosh(79/2)) - 1)/3.

%e r = 1.17455941029298007420231898869565392567594872533708249833673392030236...

%p Digits := 120: a := ((79 + 9*sqrt(77))/2)^(1/3): (a + 1/a - 1)/3: evalf(%)*10^96: ListTools:-Reverse(convert(floor(%), base, 10)); # _Peter Luschny_, Sep 15 2022

%t First[RealDigits[x/.N[First[Solve[x^3+x^2-3==0, x]], 76]]] (* _Stefano Spezia_, Sep 15 2022 *)

%o (PARI) (2*cosh((1/3)*acosh(79/2)) - 1)/3 \\ _Michel Marcus_, Sep 15 2022

%Y Cf. A356034.

%K nonn,cons,easy

%O 1,3

%A _Wolfdieter Lang_, Sep 13 2022