A199589 - OEIS

%I #27 Nov 22 2024 15:07:40

%S 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,

%T 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,

%U 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

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

%C 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.

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>.

%F Equals sqrt(4/3)*cos(Pi/18). - _Charles R Greathouse IV_, Nov 10 2011

%F Equals Product_{k>=1} (1 - (-1)^k/A016051(k)). - _Amiram Eldar_, Nov 22 2024

%e 1.13715804260325761283766795192009876258136039422990658596288796494425...

%t N[Reduce[2+6d-6d^3==0, d], 100]

%t RealDigits[(2/Sqrt[3]) * Cos[Pi/18], 10, 120][[1]] (* _Amiram Eldar_, Nov 22 2024 *)

%o (PARI) real(polroots(6*x^3-6*x-2)[3]) \\ _Charles R Greathouse IV_, Nov 10 2011

%o (PARI) polrootsreal(6*x^3-6*x-2)[3] \\ _Charles R Greathouse IV_, Apr 14 2014

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

%K nonn,cons

%O 1,3

%A _Frank M Jackson_, Nov 08 2011