A192918 Decimal expansion of the real root of r^3 + r^2 + r - 1.

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

Offset: 0

Author

Frank M Jackson, Aug 26 2011

Keywords

• nonn
• cons

Comments

The real solution r of the cubic equation r^3 + r^2 + r - 1 = 0 is the reciprocal of the tribonacci constant A058265. If the four sides of a quadrilateral form a geometric progression 1:r:r^2:r^3 where r is the common ratio then r is limited to the range 1/t < r < t where t is the tribonacci constant. More generally if f(n) is the n-th step Fibonacci constant then a polygon of n+1 sides can have sides in a geometric progression 1:r:r^2:...:r^n if the common ratio r is limited to the range 1/f(n) < r < f(n).

From Wolfdieter Lang, Aug 22 2022: (Start)

The roots of this cubic are obtained from the roots of y^3 + (2/3)*y - 34/27 after subtracting 1/3. The y-roots are y1 = (u_p^(1/3) + u_m^(1/3)*e_m)/3, y2 = (e_m*u_p^(1/3) + u_m^(1/3))/3 and y3 = e_p*(u_p^(1/3) + u_m^(1/3))/3. Here u_p = 17 + 3*sqrt(33), u_m = 17 - 3*sqrt(33), e_p = -(1 + sqrt(3)*i) and e_m = -(1 - sqrt(3)*i), where i = sqrt(-1).

The roots of the x-cubic are then x1, the present real solution, and x2 = y2 - 1/3 = -0.771844506... + 1.11514250...*i and the complex conjugate x3 = y3 - 1/3. (End)

Examples

			0.543689012692076361570855971801747986525203297650983935240...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Joerg Arndt, Matters Computational (The Fxtbook), section 38.9, A function encoding the Hilbert curve, page 748, y_1.
• Wikipedia, Triangle whose sides form a geometric progression.
• Index entries for algebraic numbers, degree 3

Crossrefs

Reciprocal of A058265.

Cf. A376841.

Programs

Magma

SetDefaultRealField(RealField(100)); (1/3)*(-1 -2/(17 +3*Sqrt(33))^(1/3) +(17+3*Sqrt(33))^(1/3)); // G. C. Greubel, Feb 06 2019

Mathematica

N[Reduce[r+r^2+r^3==1, r], 100]
RealDigits[(1/3)*(-1 -2/(17+3*Sqrt[33])^(1/3) +(17+3*Sqrt[33])^(1/3)), 10, 100][[1]] (* G. C. Greubel, Feb 06 2019 *)
RealDigits[Root[r^3+r^2+r-1,1],10,120][[1]] (* Harvey P. Dale, May 18 2023 *)

PARI

polrootsreal(r^3 + r^2 + r - 1)[1] \\ Charles R Greathouse IV, Apr 14 2014

Sage

numerical_approx((1/3)*(-1 -2/(17+3*sqrt(33))^(1/3) +(17+ 3*sqrt(33))^(1/3)), digits=100) # G. C. Greubel, Feb 06 2019

Formula

Equals (1/3)*(-1-2/(17+3*sqrt(33))^(1/3) + (17+3*sqrt(33))^(1/3)).

Equals (1/3)*(u_p^(1/3) + u_m^(1/3)*e_m - 1), with u_p = 17 + 3*sqrt(33), u_m = 17 - 3*sqrt(33), and e_m = -(1 - sqrt(3)*i), with i = sqrt(-1). - Wolfdieter Lang, Aug 22 2022

Equals hypergeom([1/4,1/2,3/4],[2/3,4/3],16/27)/2. - Gerry Martens, Jul 13 2023