A356034 - OEIS

%I #19 Nov 09 2022 04:39:07

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

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

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

%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 - 83/27.

%C The other two roots are (w1*(83/2 + (9/2)*sqrt(85))^(1/3) + w2*(83/2 - (9/2)*sqrt(85))^(1/3) + 1)/3 = -.43185326... + 1.19297873...*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 With hyperbolic function these roots are (1 - cosh((1/3)*arccosh(83/2)) + sinh((1/3)*arccosh(83/2))*sqrt(3)*i)/3, and its complex conjugate.

%F r = ((332 + 36*sqrt(85))^(1/3) + 4/(332 + 36*sqrt(85))^(1/3) + 2)/6.

%F r = ((83/2 + (9/2)*sqrt(85))^(1/3) + (83/2 - (9/2)*sqrt(85))^(1/3) + 1)/3.

%F r = (2*cosh((1/3)*arccosh(83/2)) + 1)/3.

%e 1.8637065278191890932414679152703590042315488427041530200445580733474928267...

%p Digits := 120: (332 + 36*sqrt(85))^(1/3)/2: (a + 1/a + 1)/3: evalf(%)*10^90:

%p 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]], 91]]] (* _Stefano Spezia_, Sep 15 2022 *)

%o (PARI) (2*cosh((1/3)*acosh(83/2)) + 1)/3 \\ _Michel Marcus_, Sep 15 2022

%Y Cf. A357100.

%K nonn,cons,easy

%O 1,2

%A _Wolfdieter Lang_, Sep 13 2022