A294644 Decimal expansion of the real positive solution to x^3 = x + 3.

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

Offset

1,2

Comments

From Wolfdieter Lang, Sep 13 2022: (Start)

The other two solutions are (w1*(81/2 + (3/2)*sqrt(717))^(1/3) + w2*(81/2 - (3/2)*sqrt(717))^(1/3))/3 = -0.835849940... + 1.04686931...*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 functions these solutions are -(1/3)*sqrt(3)*(cosh((1/3)*arccosh((9/2)*sqrt(3))) - sqrt(3)*sinh((1/3)*arccosh((9/2)*sqrt(3)))*i), and its complex conjugate.(End)

Links

Table of n, a(n) for n=1..105.

Index entries for algebraic numbers, degree 3

Formula

Equals (3 + (3 + (3 + ...)^(1/3))^(1/3))^(1/3).

Equals (3/2 + sqrt(239/108))^(1/3) + (3/2 - sqrt(239/108))^(1/3).

From Wolfdieter Lang, Sep 13 2022: (Start)

Equals (1/6)*(324 + 12*sqrt(717))^(1/3) + 2/(324 + 12*sqrt(717))^(1/3).

Equals ((81/2 + (3/2)*sqrt(717))^(1/3) + (81/2 - (3/2)*sqrt(717))^(1/3))/3.

Equals (2/3)*sqrt(3)*cosh((1/3)*arccosh((9/2)*sqrt(3))). (End)

Example

1.6716998816571609697481497812195572287...

Mathematica

RealDigits[x /. FindRoot[x^3 - x - 3, {x, 1}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, Jun 01 2021 *)

Prog

(PARI) solve(n=0, 2, n^3-n-3)
(PARI) polrootsreal(x^3-x-3)[1] \\ Charles R Greathouse IV, Nov 05 2017
(MATLAB) solve('x^3=x+3'); ans(1), eval(ans),  %  Michal Paulovic, Mar 08 2021

Crossrefs

Cf. A337569.

Sequence in context: A296459 A354684 A308915 * A256128 A280501 A244818

Adjacent sequences: A294641 A294642 A294643 * A294645 A294646 A294647

Keyword

nonn,cons

Author

Michal Paulovic, Nov 05 2017

Status

approved