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.

A083124 Continued fraction expansion of tanh(Pi/2).

Original entry on oeis.org

0, 1, 11, 14, 4, 1, 1, 1, 3, 1, 295, 4, 4, 1, 5, 17, 2, 8, 2, 15, 3, 1, 1, 1, 5, 54, 4, 1, 2, 1, 1, 16, 2, 2, 2, 5, 1, 1, 2, 1, 82, 1, 6, 1, 1, 1, 1, 3, 1, 1, 4, 1, 3, 3, 1, 5, 1, 1, 1, 282, 1, 5, 1, 1, 1, 1, 2, 10, 2, 1, 39, 1, 1, 5, 2, 1, 6, 4, 1, 22, 1, 1, 6, 1, 3, 5, 3, 1, 2, 9, 1, 3, 6, 23, 1, 1, 1, 14, 2
Offset: 0

Views

Author

N. J. A. Sloane, Jun 02 2003

Keywords

References

  • J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 11.

Links

Crossrefs

Cf. A060402, A084304, A367960 (decimal expansion).

Programs

  • Maple
    with(numtheory): c := cfrac (tanh(Pi/2),300,'quotients');
  • Mathematica
    ContinuedFraction[Tanh[Pi/2], 100] (* Paolo Xausa, Jul 04 2024 *)
  • PARI
    contfrac(tanh(Pi/2)) \\ Michel Marcus, Apr 11 2021

A084305 Continued fraction expansion of tanh(tanh(1)).

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 5, 3, 1, 2, 85, 1, 2, 2, 2, 1, 1, 1, 2, 5, 1, 7, 1, 1, 4, 1, 4, 1, 1, 1, 1, 6, 3, 1, 3, 3, 2, 1, 9, 12, 16, 4, 1, 1, 1, 2, 1, 2, 1, 49, 3, 7, 1, 2, 6, 1, 1, 1, 6, 1, 3, 2, 1, 1, 5, 2, 1, 199, 8, 1, 8, 1, 1, 2, 2, 1, 9, 1, 4, 4, 4, 2, 2, 1, 2, 1, 3, 1, 249, 1, 2, 2, 1, 1, 4, 1, 1, 135, 3, 2, 1
Offset: 0

Views

Author

Labos Elemer, Jun 04 2003

Keywords

Comments

While the continued fraction of tanh(1) = {0, 1, 3, 5, 7, 11, ..}, i.e., 0 and odd numbers, twice applying tanh() we get this sequence.

Links

Crossrefs

Cf. A083124, A084304.

Programs

  • Maple
    with(numtheory); cfrac(tanh(tanh(tanh(1))),300,'quotients');
  • Mathematica
    ContinuedFraction[Tanh[Tanh[1]],120] (* Harvey P. Dale, Jan 19 2024 *)
  • PARI
    contfrac(tanh(tanh(1))) \\ Amiram Eldar, Mar 08 2025

A133658 Decimal expansion of Sum_{x=integer, -inf < x < inf} (1/sqrt(2*Pi))*exp(-x^2/2).

Original entry on oeis.org

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

Views

Author

Martin Raab, Dec 28 2007

Keywords

Comments

Standard normal distribution taken at all integers x from -infinity to +infinity.
Not only is this constant quite close to 1/tanh(pi^2) (difference is about 1.43*10^-17), but it is even closer if the second term of its continued fraction, 186895766.612113..., is reduced by 1/2 (the difference then decreases to about 10^-34).
The continued fraction begins: 1, 186895766, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 6, 1, 5, 8, 1, 1, 3, 1, 44, 3, 7, 31, 2, 5, 1, 1, 5, 1, 5, 5334, 1, ... - Robert G. Wilson v, Dec 30 2007
See A084304 for cont.frac.(1/tanh(pi^2)) = [1, 186895766, 8, 1, 11, 2, 3, ...] - M. F. Hasler, Oct 24 2009

Examples

			1.000000005350575982148479362482248...

Links

Programs

  • Mathematica
    RealDigits[(1 + 2*Sum[ Exp[ -x^2/2], {x, 1, 24, 1}])/Sqrt[2 Pi], 10, 2^7][[1]] (* Robert G. Wilson v, Dec 30 2007 *)
  • PARI
    default(realprecision,100); sqrt(2/Pi)*(suminf(k=1,exp(-k^2/2))+.5)
vecextract(eval(Vec(Str( % ))),"^2") \\ M. F. Hasler, Oct 24 2009

Extensions

More terms from Robert G. Wilson v, Dec 30 2007
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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