OFFSET
1,2
COMMENTS
This equals r0 + 1/3 where r0 is the real root of y^3 - (1/3)*y - 83/27.
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.
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.
FORMULA
r = ((332 + 36*sqrt(85))^(1/3) + 4/(332 + 36*sqrt(85))^(1/3) + 2)/6.
r = ((83/2 + (9/2)*sqrt(85))^(1/3) + (83/2 - (9/2)*sqrt(85))^(1/3) + 1)/3.
r = (2*cosh((1/3)*arccosh(83/2)) + 1)/3.
EXAMPLE
1.8637065278191890932414679152703590042315488427041530200445580733474928267...
MAPLE
Digits := 120: (332 + 36*sqrt(85))^(1/3)/2: (a + 1/a + 1)/3: evalf(%)*10^90:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 15 2022
MATHEMATICA
First[RealDigits[x/.N[First[Solve[x^3-x^2-3==0, x]], 91]]] (* Stefano Spezia, Sep 15 2022 *)
PROG
(PARI) (2*cosh((1/3)*acosh(83/2)) + 1)/3 \\ Michel Marcus, Sep 15 2022
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 13 2022
STATUS
approved