A117605 Decimal expansion of the real solution to equation x^3 + 3*x = 2.

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

Offset

0,1

Comments

A014176 Decimal expansion of the silver mean, 1+sqrt(2). Interestingly, when we look for x=0.596071637983321523112805414399681828113325, the original Inverse Symbolic Calculator replies: "... Roots of polynomials of 4th degree (coeffs: -9..9) 5960716379833215 = 10+15*x+5*x^3" - with wrong sign of the first term.

Let p(n,x) denote the n-th Maclaurin polynomial of e^x, and let p'(n,x) denote its derivative. Then p'(n+1,x) = p(n,x), so that the real zero r of p(n,x), for odd n, is also the value of x that minimizes p(n+1,x). Let y = 0.5960716... . Then for n = 3, we have r = - y - 1; see A332324 for the minimal value of p(4,x). - Clark Kimberling, Feb 13 2020

Links

Table of n, a(n) for n=0..99.

Original Inverse Symbolic Calculator

Formula

x = (1+sqrt(2))^(1/3) - 1/(1+sqrt(2))^(1/3).

Example

x = 0.596071637983321523112805414399681828113325...

Mathematica

RealDigits[N[Solve[3*z+z^3==2, z][[1, 1, 2]], 100]][[1]]

Crossrefs

Cf. A014176, A332324.

Sequence in context: A133742 A134879 A051158 * A303983 A073003 A344093

Adjacent sequences: A117602 A117603 A117604 * A117606 A117607 A117608

Keyword

cons,nonn

Author

Zak Seidov, Apr 27 2006

Extensions

Website URL and name (changed from "Plouffe's Inverter") updated by Jon E. Schoenfield, Feb 28 2020

a(99) corrected by Sean A. Irvine, Jul 25 2021

Status

approved