A199590 Decimal expansion (unsigned) of the greatest root of 6x^3 + 18x^2 + 12x + 2 = 0.

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

Offset

0,1

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 f = -0.257772801... and is the greatest root of the equation: 2 + 12d + 18d^2 + 6d^3 = 0. The value of g is given in A199589.

Links

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

Index entries for algebraic numbers, degree 3

Formula

sqrt(4/3)*sin(Pi*2/9) - 1. - Charles R Greathouse IV, Nov 10 2011

Example

-0.257772801031440844729449397270635822708944125700977319782314639395808...

Mathematica

N[Reduce[2+12d+18d^2+6d^3==0, d], 100]

Prog

(PARI) real(polroots(6*x^3+18*x^2+12*x+2)[3]) \\ Charles R Greathouse IV, Nov 10 2011
(PARI) polrootsreal(6*x^3-18*x^2+12*x-2)[1] \\ Charles R Greathouse IV, Oct 27 2023

Crossrefs

Cf. A010503, A199220, A199221, A199589.

Sequence in context: A373919 A131688 A226213 * A096624 A145378 A069887

Adjacent sequences: A199587 A199588 A199589 * A199591 A199592 A199593

Keyword

nonn,cons

Author

Frank M Jackson, Nov 08 2011

Extensions

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

Status

approved