A199589 Decimal expansion of the greatest root of 6x^3 - 6x - 2 = 0.

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

Offset: 1

Author

Frank M Jackson, Nov 08 2011

Keywords

• nonn
• cons

Comments

If the side lengths of a quadrilateral form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) where d is the common difference between the denominators of the harmonic progression, then the triangle inequality condition requires that d be in the range f < d < g, where g = 1.1371580... and is the greatest root of the equation: 2 + 6d - 6d^3 = 0. The value of f is given in A199590.

Examples

			1.13715804260325761283766795192009876258136039422990658596288796494425...
		

Links

• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A010503, A016051, A199220, A199221, A199590.

Programs

Mathematica

N[Reduce[2+6d-6d^3==0, d], 100]
RealDigits[(2/Sqrt[3]) * Cos[Pi/18], 10, 120][[1]] (* Amiram Eldar, Nov 22 2024 *)

PARI

real(polroots(6*x^3-6*x-2)[3]) \\ Charles R Greathouse IV, Nov 10 2011

PARI

polrootsreal(6*x^3-6*x-2)[3] \\ Charles R Greathouse IV, Apr 14 2014

Formula

Equals sqrt(4/3)*cos(Pi/18). - Charles R Greathouse IV, Nov 10 2011

Equals Product_{k>=1} (1 - (-1)^k/A016051(k)). - Amiram Eldar, Nov 22 2024