A356034 Decimal expansion of the real root of x^3 - x^2 - 3.

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

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.

Links

Table of n, a(n) for n=1..91.

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

Cf. A357100.

Sequence in context: A085666 A084895 A154719 * A263181 A298978 A199151

Adjacent sequences: A356031 A356032 A356033 * A356035 A356036 A356037

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Sep 13 2022

Status

approved