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 1-5 of 5 results.

A073743 Decimal expansion of cosh(1).

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Aug 07 2002

Keywords

Comments

Also decimal expansion of cos(i). - N. J. A. Sloane, Feb 12 2010
cosh(x) = (e^x + e^(-x))/2.
Equals Sum_{n>=0} 1/A010050(n). See Gradsteyn-Ryzhik (0.245.5). - R. J. Mathar, Oct 27 2012
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

Examples

			1.54308063481524377847790562075...

References

  • S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:6 at page 20.

Links

Crossrefs

Cf. A068118 (continued fraction), A073742, A073744, A073745, A073746, A073747, A049470, A137204.

Programs

  • Maple
    Digits:=100: evalf(cosh(1)); # Wesley Ivan Hurt, Nov 18 2014
  • Mathematica
    RealDigits[Cosh[1],10,120][[1]] (* Harvey P. Dale, Aug 03 2014 *)
  • PARI
    cosh(1)

Formula

Continued fraction representation: cosh(1) = 1 + 1/(2 - 2/(13 - 12/(31 - ... - (2*n - 4)*(2*n - 5)/((4*n^2 - 10*n + 7) - ... )))). See A051396 for proof. Cf. A049470 (cos(1)) and A073742 (sinh(1)). - Peter Bala, Sep 05 2016
Equals Product_{k>=0} 1 + 4/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073746 = A137204/2. - Hugo Pfoertner, Dec 27 2024

A073746 Decimal expansion of sech(1).

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Aug 07 2002

Keywords

Comments

sech(x) = 2/(e^x + e^(-x)).
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

Examples

			0.64805427366388539957497735322...

References

  • Samuel M. Selby (ed.), CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Links

Crossrefs

Cf. A068118 (continued fraction), A073743 (cosh(1)=1/A073746), A073742 (sinh(1)), A073744 (tanh(1)), A073745 (csch(1)), A073747 (coth(1)), A122045.

Programs

  • Mathematica
    RealDigits[Sech[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)
  • PARI
    1/cosh(1)

Formula

Equals Sum_{k>=0} E(2*k) / (2*k)!, where E(k) is the k-th Euler number (A122045). - Amiram Eldar, May 15 2021

A068139 Continued fraction expansion for sinh(1).

Original entry on oeis.org

1, 5, 1, 2, 2, 2, 1, 2, 7, 5, 1, 1, 1, 2, 2, 19, 1, 2, 1, 7, 1, 1, 9, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 4, 5, 3, 5, 1, 3, 1, 1, 1, 2, 7, 1, 9, 1, 1, 2, 1, 21, 1, 18, 1, 2, 734, 1, 2, 1, 84, 2, 2, 1, 1, 2, 10, 1, 3, 7, 1, 16, 1, 2, 4, 56, 1, 13, 16, 208
Offset: 0

Views

Author

Benoit Cloitre, Mar 13 2002

Keywords

Comments

If an extra zero is added to the beginning of this sequence, continued fraction for csch(1) = 1/sinh(1). - Rick L. Shepherd, Aug 07 2002

Links

Crossrefs

Cf. A068118, A073742 (decimal expansion), A073745 (decimal expansion of csch(1)).
Cf. A078980, A078981 (convergents).

Programs

Extensions

Offset changed by Andrew Howroyd, Aug 05 2024

A078982 Denominators of continued fraction convergents to cosh(1).

Original entry on oeis.org

1, 1, 2, 11, 35, 116, 267, 383, 8310, 8693, 17003, 25696, 119787, 265270, 385057, 18748006, 1331493483, 9339202387, 10670695870, 20009898257, 190759780183, 210769678440, 612299137063, 823068815503, 9666056107596, 10489124923099
Offset: 0

Views

Author

Benoit Cloitre, Dec 20 2002

Keywords

Links

Crossrefs

Cf. A068118 (continued fraction), A073743 (decimal expansion), A078983 (numerators).

Programs

  • Mathematica
    Denominator[Convergents[Cosh[1],30]] (* Harvey P. Dale, Jul 22 2014 *)
  • PARI
    a(n)=component(component(contfracpnqn(contfrac(cosh(1),n+1)),1),2)

Extensions

Offset changed by Andrew Howroyd, Aug 05 2024

A078983 Numerators of continued fraction convergents to cosh(1).

Original entry on oeis.org

1, 2, 3, 17, 54, 179, 412, 591, 12823, 13414, 26237, 39651, 184841, 409333, 594174, 28929685, 2054601809, 14411142348, 16465744157, 30876886505, 294357722702, 325234609207, 944826941116, 1270061550323, 14915503994669, 16185565544992
Offset: 0

Views

Author

Benoit Cloitre, Dec 20 2002

Keywords

Links

Crossrefs

Cf. A068118 (continued fraction), A073743 (decimal expansion), A078982 (denominators).

Programs

  • Mathematica
    Numerator[Convergents[Cosh[1],30]] (* Harvey P. Dale, Feb 02 2012 *)
  • PARI
    a(n)=component(component(contfracpnqn(contfrac(cosh(1),n+1)),1),1)

Extensions

Offset changed by Andrew Howroyd, Aug 05 2024
Showing 1-5 of 5 results.

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

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