A109134 Decimal expansion of Phi, the real root of the equation 1/x = (x-1)^2.

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

Offset: 1

Author

Lekraj Beedassy, Aug 17 2005

Keywords

• cons
• nonn

Comments

The silver number (A060006) is equal to Phi*(Phi-1).

Also Phi*(Phi-1) = 1/(Phi-1). - Richard R. Forberg, Oct 08 2014

Equations to which this is a root can also be written as: x = sqrt(x + sqrt(x)); x^2 - x - sqrt(x) = 0; or this form where n = 1: x = n + 1/sqrt(x). When n = 2 then the root is 2.618033988... = A104457 = 1 + A001622 or 1 + "Golden Ratio" called phi. - Richard R. Forberg, Oct 08 2014

Also equals the largest root (negated) of the Mandelbrot polynomial P_2(z) = 1+z*(1+z)^2. - Jean-François Alcover, Apr 16 2015

Suppose that r is a real number in the interval [3/2, 5/3). Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k). Conjectures: the limit L(r) of c(k+1)/c(k) as k -> oo exists, L(r) is discontinuous at 5/3 (cf. A279676), and the left limit of L(r) as r->5/3 is Phi. - Clark Kimberling, Jul 11 2017

From Wolfdieter Lang, Nov 07 2022: (Start)

This equals r + 2/3 where r is the real root of y^3 - (1/3)*y - 25/27.

The other roots of x^3 - 2*x^2 + x - 1 are (2 + w1*((25 + 3*sqrt(69))/2)^(1/3) + w2*((25 - 3*sqrt(69))/2)^(1/3))/3 = 0.1225611668... + 0.7448617668...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex conjugate roots of x^3 - 1.

Using hyperbolic functions these roots are (2 - cosh((1/3)*arccosh(25/2)) + sqrt(3)*sinh((1/3)*arccosh(25/2))*i)/3, and its complex conjugate. (End)

Examples

			1.75487766624669276004950889635852869189460661777279314398928397064...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.11, p. 340.
• Martin Gardner, A Gardner's Workout, pp. 124-126, A. K. Peters MA 2001.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Simon Baker, On small bases which admit countably many expansions, Journal of Number Theory, Volume 147, February 2015, Pages 515-532.
• Simon Plouffe, Plouffe's Inverter .
• Nikita Sidorov, Expansions in non-integer bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Prop. 2.3 p. 744.
• Yuru Zou and Derong Kong, On a problem of countable expansions, Journal of Number Theory, Volume 158, January 2016, Pages 134-150. See Theorem 1.1 p. 135.
• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A001622, A001685, A060006, A075778, A104457, A358181.

Programs

Mathematica

FindRoot[x^3 - 2x^2 + x - 1 == 0, {x, 1.75}, WorkingPrecision -> 128][[1, 2]] (* Robert G. Wilson v, Aug 19 2005 *)
Root[x^3-2x^2+x-1, x, 1] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 05 2013 *)

PARI

d=104;default(realprecision,d);print(k=solve(x=1,2,(x-1)^2-1/x)); for(c=0,d,z=floor(k);print1(z,",",);k=10*(k-z))

PARI

polrootsreal(x^3-2*x^2+x-1)[1] \\ Charles R Greathouse IV, Aug 15 2014

Formula

Equals 1+A075778. - R. J. Mathar, Aug 20 2008

Equals (1/6*(108+12*sqrt(69))^(1/3) + 2/(108+12*sqrt(69))^(1/3))^2. - Vaclav Kotesovec, Oct 08 2014

Equals Rho^2 where Rho is the plastic number 1.3247179572...(see A060006). - Philippe Deléham, Sep 29 2020

From Wolfdieter Lang, Nov 07 2022: (Start)

Equals (2 + ((25 + 3*sqrt(69))/2)^(1/3) + ((25 + 3*sqrt(69))/2)^(-1/3))/3.

Equals (2 + ((25 + 3*sqrt(69))/2)^(1/3) + ((25 - 3*sqrt(69))/2)^(1/3))/3.

Equals 2*(1 + cosh((1/3)*arccosh(25/2)))/3. (End)

Equals - Sum_{k>=1} Gamma(k - k/5 - 1)*Gamma(k/5 + 1)*sin(3*k*Pi/5)/(k*Pi*Gamma(k)). - Antonio Graciá Llorente, Dec 14 2024

Extensions

Extended by Klaus Brockhaus and Robert G. Wilson v, Aug 19 2005