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 1-2 of 2 results.

A060007 Decimal expansion of the positive real root of x^4 - x - 1.

Original entry on oeis.org

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

Views

Author

Fabian Rothelius, Mar 14 2001

Keywords

Comments

Original name: Decimal expansion of v_4, where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1.
v_2 = A001622 - 1. [Corrected by M. F. Hasler, Jul 12 2025]
v_3 = A060006, a.k.a. plastic constant, real root of x^3 - x - 1. - M. F. Hasler, Jul 12 2025
A Perron number of the 4th degree polynomial (see Boys and Wu). - R. J. Mathar, Mar 19 2011
This number is not ruler-and-compass constructible because x^4-x-1 and its resolvent x^3+4x+1 are irreducible over the rationals. - Jean-François Alcover, Aug 31 2015
The other (negative) real root -0.724491959... is -A356032. The first of the pair of complex conjugate roots is obtained by negating in the formula for v_4 below sqrt(2*u) and sqrt(u), giving -0.2481260628... - 1.0339820609...*i. - Wolfdieter Lang, Aug 27 2022
The sequence a(n) = v_4^((n^2-n)/2) satisfies the Somos-4 recursion a(n+2)*a(n-2) = a(n+1)*a(n-1) + a(n)^2 for all n in Z. - Michael Somos, Mar 24 2023

Examples

			v_4 = 1.220744084605759475361685349...

Links

Crossrefs

Cf. A001622 (golden ratio, root of x^2 - x - 1), A060006 (plastic number, root of x^3 - x - 1), A202540 (log thereof), A160155 (root of x^5 - x - 1), A356032 (root of x^4 + x - 1), A006720, A298813.

Programs

  • Maple
    r:=(108+12*sqrt(849))^(1/3): (sqrt(12/sqrt(-8/r+r/6)+48/r-r) + sqrt(-48/r+r))/(2*sqrt(6)): evalf(%,105); # Vaclav Kotesovec, Oct 12 2013
  • Mathematica
    RealDigits[x/.FindRoot[x^4==x+1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jul 11 2012 *)
Root[ #^4 - # - 1&, 2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 04 2013 *)
  • PARI
    default(realprecision, 110); digits(floor(solve(x=1, 2, x^4 - x - 1)*10^105)) /* Michael Somos, Mar 22 2023 */

Formula

Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/4))^(1/4))^(1/4))^(1/4))^(1/4). - Ilya Gutkovskiy, Dec 15 2017
v_4 = (sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)), ep = (-1 + sqrt(3)*i)/2 and i = sqrt(-1).
For the trigonometric equivalent u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))). - Wolfdieter Lang, Aug 27 2022
Equals 1 + Sum_{n >= 1} (1/4)^n*(Product_{j=1..n-1} 1 + n - 4*j)/n!. - Antonio Graciá Llorente, Dec 13 2024
Equals exp(A202540) = sqrt(A298813). - Hugo Pfoertner, Dec 14 2024

Extensions

More terms from Benoit Cloitre, Jan 11 2003
Simplified definition from M. F. Hasler, Jul 12 2025

A202537 Decimal expansion of x satisfying e^x-e^(-2x)=1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 21 2011

Keywords

Comments

If u>0 and v>0, there is a unique number x satisfying e^(ux)-e^(-vx)=1. Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A002390
1.... 2.... A202537
1.... 3.... A202538
2.... 1.... A202539
3.... 1.... A202540
2.... 2.... A202541
3.... 3.... A202542
1/2..1/2... A202543
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z=g(u,v) an implicit surface of f. For an example related to A202537, take f(x,u,v)=e^(ux)-e^(-vx)-1 and g(u,v) = a nonzero solution x of f(x,u,v)=0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

Examples

			0.382245085840035641329358499184857393759416422...

Links

Crossrefs

Cf. A002390.

Programs

  • Mathematica
    (* Program 1:  A202537 *)
u = 1; v = 2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .3, .4}, WorkingPrecision -> 110]
RealDigits[r]  (* A202537 *)
(* Program 2: implicit surface for e^(ux)-e(-vx)=1 *)
f[{x_, u_, v_}] := E^(u*x) - E^(-v*x) - 1;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, 1, 4}, {u, 2, 20}];
ListPlot3D[Flatten[t, 1]] (* for A202537 *)
First[ RealDigits[ Log[ Root[#^3 - #^2 - 1 & , 1]], 10, 99]] (* Jean-François Alcover, Feb 26 2013 *)
  • PARI
    solve(x=0,1,exp(x)-exp(-2*x)-1) \\ Charles R Greathouse IV, Feb 26 2013
  • PARI
    log(polrootsreal(x^3-x^2-1)[1]) \\ Charles R Greathouse IV, Feb 07 2025

Extensions

Digits from a(90) on corrected by Jean-François Alcover, Feb 26 2013
Showing 1-2 of 2 results.

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