cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Lekraj Beedassy, Aug 17 2005

Keywords

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

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

A088559 Decimal expansion of R^2 where R^2 is the real root of x^3 + 2*x^2 + x - 1 = 0.

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Nov 19 2003

Keywords

Comments

Arise in a study of AGM (arithmetic-geometric mean) and HGM (harmonic-geometric mean) - like sequences. Let u(k+1)=sqrt(u(k)*v(k)); v(k+1)=v(k)+u(k) and r(k+1)=sqrt(r(k)*s(k)); s(k+1)=1/(1/r(k)+1/s(k)). Then for any positive initial values u(0),v(0),r(0),s(0) limit k-->oo u(k)/v(k)= limit k-->oo s(k)/r(k)=R^2.
From Wolfdieter Lang, Nov 07 2022: (Start)
This equals r0 - 2/3 where r0 is the real root of y^3 - (1/3)*y - 29/27.
The other roots of x^3 + 2*x^2 + x - 1 are (-2 + w1*((29 + 3*sqrt(93))/2)^(1/3) + w2*((29 - 3*sqrt(93))/2)^(1/3))/3 = -1.2327856159... + 0.7925519925...*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(29/2)) + sqrt(3)*sinh((1/3)*arccosh(29/2))*i)/3, and its complex conjugate. (End)

Examples

			0.465571231876768026656731225219939108025577568472285701643183111249262996685...

Links

Crossrefs

Cf. A092526, A144229, A358181.

Programs

  • Mathematica
    Root[x^3 + 2x^2 + x - 1, 1] // RealDigits[#, 10, 104]& // First (* Jean-François Alcover, Mar 04 2013 *)
  • PARI
    allocatemem(932245000); default(realprecision, 20080); x=10*solve(x=0, 1, x^3 + 2*x^2 + x - 1); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b088559.txt", n, " ", d)); \\ Harry J. Smith, Jun 21 2009
  • PARI
    polrootsreal(x^3 + 2*x^2 + x - 1)[1] \\ Charles R Greathouse IV, Mar 03 2016

Formula

R^2=0.46557123187676... 1+R^2=1.46557123187676... = A092526 constant.
From Vaclav Kotesovec, Dec 18 2014: (Start)
Equals (1/6)*(116+12*sqrt(93))^(1/3) + 2/(3*(116+12*sqrt(93))^(1/3)) - 2/3.
Equals 2*cos(arccos(29/2)/3)/3 - 2/3.
Equals A092526 - 1.
(End)
From Wolfdieter Lang, Nov 07 2022: (Start)
Equals (-2 + ((29 + 3*sqrt(93))/2)^(1/3) + ((29 + 3*sqrt(93))/2)^(-1/3))/3.
Equals (-2 + ((29 + 3*sqrt(93))/2)^(1/3) + ((29 - 3*sqrt(93))/2)^(1/3))/3.
Also with hperbolic cosh and arccosh instead of cos and arccos above.
(End)

A358182 Decimal expansion of the real root of 2*x^3 - x^2 - x - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Nov 07 2022

Keywords

Comments

This equals r0 + 1/6 where r0 is the real root of y^3 - (7/12)*y - 16/27.
The other roots of 2*x^3 - x^2 - x - 1 are (1 + w1*(64 + 3*sqrt(417))^(1/3) + w2*(64 - 3*sqrt(417))^(1/3))/6 = -0.3668759642... + 0.5202594388...*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 (1 - sqrt(7)*( cosh((1/3)*arccosh((64/49)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((64/49)*sqrt(7)))*i))/6, and its complex conjugate.

Examples

			1.23375192852825878819094337767939303519112723753118649423200988702753759...

Crossrefs

Cf. A358181, A358183.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[2*x^3 - x^2 - x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)

Formula

r = (1 + (64 + 3*sqrt(417))^(1/3) + 7*(64 + 3*sqrt(417))^(-1/3))/6.
r = (1 + (64 + 3*sqrt(417))^(1/3) + (64 - 3*sqrt(417))^(1/3))/6.
r = (1 + 2*sqrt(7)*cosh((1/3)*arccosh((64/49)*sqrt(7))))/6.
r = (1/6) + (4/3)*Hyper2F1([-1/6,1/3],[1/2],3753/4096). - Gerry Martens, Nov 08 2022
Showing 1-3 of 3 results.

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