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.

A085846 Decimal expansion of root of x = (1+1/x)^x.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jul 05 2003

Keywords

Comments

Equivalently, the root of x^(x+1) = (x+1)^x.
Also a root of 1/(x^(1/x)-1) - x = 0 and 1/(x^(1/x)-1/x-1) - x = 0, which also contains the root 5.50798565277317825758902... 1/(x^(1/x)-1) ~ Pi(x) and 1/(x^(1/x)-1/x-1) ~ Pi(x), which is a much better approximation. These roots also can be computed by the recurrences x = 1/(x^(1/x)-1) and x = 1/(x^(1/x)-1/x-1). - Cino Hilliard, Sep 13 2008
This constant is transcendental (Lord, 2002). - Amiram Eldar, Oct 29 2022

Examples

			2.2931662874118610315080282912508058643722572903271212485377103961...

Links

Crossrefs

Cf. A021002, A124930, A169862.

Programs

  • Mathematica
    RealDigits[ FindRoot[x^(1/x) - (x + 1)^(1/(x + 1)) == 0, {x, 2}, WorkingPrecision -> 128][[1, 2]], 10, 111][[1]] (* Robert G. Wilson v *)
  • PARI
    solve(x=2,3,(1+1/x)^x-x) \\ Charles R Greathouse IV, Apr 14 2014

Formula

x satisfies x^(1/x) = (x+1)^(1/(x+1)). - Marco Matosic, Nov 25 2005

A356562 Decimal expansion of the unique positive real root of the equation x^x^x = x^x + 1.

Original entry on oeis.org

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

Views

Author

Marco Ripà, Aug 12 2022

Keywords

Comments

This constant is case m=2 in a family of real roots of x^^(m+1) - x^^m = 1, where ^^ is tetration. These roots have lim_{m->inf} x(m) = e^(1/e) (see A073229).

Examples

			1.67129219798893255...

Crossrefs

Cf. A073229, A124930 (m=1).

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^(x^x) == x^x + 1, {x, 2}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Aug 13 2022 *)
  • PARI
    solve(x=1, 2, x^x^x - x^x - 1) \\ Michel Marcus, Aug 13 2022

A356805 Decimal expansion of the unique positive real root of the equation x^x^(x - 1) = x + 1.

Original entry on oeis.org

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

Views

Author

Marco Ripà and Flavio Niccolò Baglioni, Aug 28 2022

Keywords

Comments

This constant arises from a well-known linear approximation for real height of the tetration x^^x (for x belonging to (1, 2)), where x^^x indicates the tetration of the real base x having the same height (see Links - Wikipedia).
A valuable method to extend tetration to real numbers, and solving equations as the above, has been introduced in 2006 by Hooshmand in his paper "Ultra power and ultra exponential functions" (see Links - Hooshmand).

Examples

			1.85566023196173...

Links

Crossrefs

Cf. A124930, A356562.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^(x^(x - 1)) == x + 1, {x, 2}, WorkingPrecision -> 100]][[1]]
  • PARI
    solve(x=1, 2, x^x^(x - 1) - x - 1) \\ Michel Marcus, Aug 29 2022
Showing 1-3 of 3 results.

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