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

Author

Zak Seidov, Apr 27 2006

Keywords

• cons
• nonn

Comments

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

Examples

			x = 0.596071637983321523112805414399681828113325...
		

Crossrefs

Cf. A014176, A332324.

Programs

Mathematica

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

Formula

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

From Gerry Martens, Mar 23 2025: (Start)

Equals (2/3)*hypergeom([1/3, 2/3], [3/2], -1).

Equals 2*sinh(asinh(1)/3). (End)

Extensions

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