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 1-10 of 14 results. Next

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.

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

A073742 Decimal expansion of sinh(1).

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Aug 07 2002

Keywords

Comments

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019
Decimal expansion of u > 0 such that 1 = arclength on the hyperbola y^2 - x^2 = 1 from (0,0) to (u,y(u)). - Clark Kimberling, Jul 04 2020

Examples

			1.17520119364380145688238185059...
		

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 2:5:7 at page 20.

Crossrefs

Cf. A068139 (continued fraction), A073743, A073744, A073745, A073746, A073747, A049469, A049470, A174548.

Programs

  • Mathematica
    First@ RealDigits@ N[Sinh@ 1, 120] (* Michael De Vlieger, Sep 04 2016 *)
  • PARI
    sinh(1)

Formula

Equals (e - e^(-1))/2.
Equals sin(i)/i. - N. J. A. Sloane, Feb 12 2010
Equals Sum_{n>=0} 1/A009445(n). See Gradsteyn-Ryzhik (0.245.6.) - R. J. Mathar, Oct 27 2012
Continued fraction representation: sinh(1) = 1 + 1/(6 - 6/(21 - 20/(43 - 42/(73 - ... - (2*n - 1)*(2*n - 2)/((2*n*(2*n + 1) + 1) - ... ))))). See A051397 for proof. Cf. A049469. - Peter Bala, Sep 02 2016
Equals Product_{k>=1} 1 + 1/(k * Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073745 = A174548/2. - Hugo Pfoertner, Dec 27 2024

A004273 0 together with odd numbers.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131
Offset: 0

Views

Author

Keywords

Comments

Also continued fraction for tanh(1) (A073744 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
From Jaroslav Krizek, May 28 2010: (Start)
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is an integer. A040001(a(n)) = 1. See A145051 and A040001.
For n >= 1, a(n) = corresponding values of antiharmonic means to numbers from A016777 (numbers k such that antiharmonic mean of the first k positive integers is an integer).
a(n) = A000330(A016777(n)) / A000217(A016777(n)) = A146535(A016777(n)+1). (End)
If the n-th prime is denoted by p(n) then it appears that a(j) = distinct, increasing values of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) for each j. - Christopher Hunt Gribble, Oct 05 2010
A214546(a(n)) > 0. - Reinhard Zumkeller, Jul 20 2012
Dimension of the space of weight 2n+2 cusp forms for Gamma_0(6).
The size of a maximal 2-degenerate graph of order n-1 (this class includes 2-trees and maximal outerplanar graphs (MOPs)). - Allan Bickle, Nov 14 2021
Numbers not considered even for the purpose of roulette. J. Lowell, Apr 29 2025

Examples

			G.f. = x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 + 11*x^6 + 13*x^7 + 15*x^8 + 17*x^9 + ...
		

Crossrefs

Cf. A110185, continued fraction expansion of 2*tanh(1/2), and A204877, continued fraction expansion of 3*tanh(1/3). [Bruno Berselli, Jan 26 2012]
Cf. A005408.

Programs

Formula

G.f.: x*(1+x)/(-1+x)^2. - R. J. Mathar, Nov 18 2007
a(n) = lodumo_2(A057427(n)). - Philippe Deléham, Apr 26 2009
Euler transform of length 2 sequence [3, -1]. - Michael Somos, Jul 03 2014
a(n) = (4*n - 1 - (-1)^(2^n))/2. - Luce ETIENNE, Jul 11 2015

A073747 Decimal expansion of coth(1).

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Aug 07 2002

Keywords

Comments

coth(x) = (e^x + e^(-x))/(e^x - e^(-x)).
Because the continued fraction for coth(1) is all positive odd numbers in sequence, the second Mathematica program below also generates the sequence. - Harvey P. Dale, Oct 15 2011
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

Examples

			1.31303528549933130363616124693...
		

References

  • Samuel M. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Crossrefs

Cf. A005408 (continued fraction: odd numbers), A073821 (continued fraction exp. is even numbers), A073744 (tanh(1)=1/A073747), A073742 (sinh(1)), A073743 (cosh(1)), A073745 (csch(1)), A073746 (sech(1)), A349004.

Programs

  • Mathematica
    RealDigits[Coth[1],10,120][[1]] (* or *) RealDigits[ FromContinuedFraction[ Range[1,1001,2]],10,120][[1]] (* Harvey P. Dale, Oct 15 2011 *) (* see Comments, above, for the second program *)
  • PARI
    1/tanh(1)

Formula

Equals 1 + Sum_{n>=1} (2^(2*n)*B(2*n))/(2*n)! = 1 + Sum_{n>=1} (-1)^(n+1)*2*(A046988(n+1) / A002432(n+1)). - Terry D. Grant, May 30 2017
Equals 1 + BesselI(3/2, 1)/BesselI(1/2, 1). - Terry D. Grant, Jun 18 2018
Equals 1 + Sum_{k>=1} csch(2^k) (Ohtsuka, 2015; Stenger, 2017). - Amiram Eldar, Oct 04 2021

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.

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

A073745 Decimal expansion of csch(1).

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Aug 07 2002

Keywords

Comments

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

Examples

			0.85091812823932154513384276328...
		

References

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

Crossrefs

Cf. A068139 (continued fraction), A073742 (sinh(1)=1/A073745), A073743 (cosh(1)), A073744 (tanh(1)), A073746 (sech(1)), A073747 (coth(1)).

Programs

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

Formula

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

A348131 a(n) is the numerator of the relativistic sum of n velocities of 1/n, in units where the speed of light is 1.

Original entry on oeis.org

1, 4, 7, 272, 211, 51012, 14197, 18640960, 1690981, 11225320100, 313968931, 10079828372880, 83828316391, 12627774819845668, 30436810578889, 21046391759976988928, 14425381885981321, 45032132922921758270916, 8649148282327007911, 120314227994702795221920400
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2021

Keywords

Examples

			The fractions begins with 1, 4/5, 7/9, 272/353, 211/275, 51012/66637, 14197/18571, 18640960/24405761, 1690981/2215269, 11225320100/14712104501, ...
For n = 2, the sum of two velocities of 1/2 is (1/2 + 1/2)/(1 + (1/2)*(1/2)) = 4/5, thus a(2) = 4.
		

Crossrefs

Cf. A073744, A348051, A348052, A348132 (denominators).

Programs

  • Mathematica
    f[n_] := Module[{s = 1/n}, Do[s = (s + 1/n)/(1 + s/n), {k, 1, n - 1}]; s]; Numerator @ Array[f, 20]

Formula

a(n)/A348132(n) = tanh(n * arctanh(1/n)).
Lim_{n->oo} a(n)/A348132(n) = tanh(1) (A073744).
a(2n-1) = n^(2n-1) - (n-1)^(2n-1) and a(2n) = ((2n+1)^(2n) - (2n-1)^(2n)) / 2. - Thomas Ordowski, Feb 12 2022

A298242 Decimal expansion of BesselI(1,1/2)/BesselI(0,1/2).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jan 15 2018

Keywords

Examples

			0.2424996125808019453507023535036354074122660448659455966...
		

Crossrefs

Programs

  • Mathematica
    RealDigits[BesselI[1, 1/2]/BesselI[0, 1/2], 10, 110] [[1]]
    RealDigits[Hypergeometric0F1[2, (1/2)^2/4]/(4 Gamma[2] Hypergeometric0F1[1, (1/2)^2/4]), 10, 110][[1]]
  • PARI
    besseli(1,1/2)/besseli(0,1/2) \\ Michel Marcus, Jul 03 2018

Formula

Equals 1/(4 + 1/(8 + 1/(12 + 1/(16 + 1/(20 + 1/(24 + ...)))))).

A308739 Decimal expansion of BesselI(1/3,2/3)/BesselI(-2/3,2/3).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jun 21 2019

Keywords

Comments

From Peter Bala, Nov 28 2019: (Start)
Denoting this constant by c, we have the related simple continued fraction expansions:
3*c = [2; 2, 2, 2, 30, 4, 2, 1, 4, 1, 2, 6, 66, 8, 2, 1, 8, 1, 2, 10, ..., 3*(12*k + 10), 4*k + 4, 2, 1, 4*k + 4, 1, 2, 4*k + 6, ...];
(1/3)*c = [0; 3, 1, 2, 1, 1, 1, 2, 3, 39, 5, 2, 1, 5, 1, 2, 7, 75, 9, 2, 1, 9, 1, 2, 11, ..., 3*(12*k + 1), 4*k + 1, 2, 1, 4*k + 1, 1, 2, 4*k + 3, ...]. (End)

Examples

			0.8054800223869180458735566274757864104391314464...
		

Crossrefs

Cf. A016777 (continued fraction), A073744, A298241, A308740, A308741, A308742, A308743, A308744.

Programs

  • Mathematica
    RealDigits[BesselI[1/3, 2/3]/BesselI[-2/3, 2/3], 10, 110] [[1]]
  • PARI
    besseli(1/3,2/3)/besseli(-2/3,2/3) \\ Felix Fröhlich, Dec 01 2019

Formula

Equals 1/(1 + 1/(4 + 1/(7 + 1/(10 + 1/(13 + 1/(16 + 1/(19 + 1/(22 + 1/(25 + 1/(28 + ...)))))))))).

A308740 Decimal expansion of BesselI(2/3,2/3)/BesselI(-1/3,2/3).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jun 21 2019

Keywords

Examples

			0.45554452608187355662518203623334796282748850507693...
		

Crossrefs

Cf. A016789 (continued fraction), A073744, A298241, A308739, A308741, A308742, A308743, A308744.

Programs

  • Mathematica
    RealDigits[BesselI[2/3, 2/3]/BesselI[-1/3, 2/3], 10, 110] [[1]]
  • PARI
    besseli(2/3,2/3)/besseli(-1/3,2/3) \\ Felix Fröhlich, Dec 01 2019

Formula

Equals 1/(2 + 1/(5 + 1/(8 + 1/(11 + 1/(14 + 1/(17 + 1/(20 + 1/(23 + 1/(26 + 1/(29 + ...)))))))))).
From Peter Bala, Nov 29 2019: (Start)
Denoting this constant by c, we have the related simple continued fraction expansions:
3*c = [1; 2, 1, 2, 1, 2, 33, 4, 1, 2, 5, 2, 1, 6, 69, 8, 1, 2, 9, 2, 1, 10, ..., 3*(12*k + 11), 4*k + 4, 1, 2, 4*k + 5, 2, 1, 4*k + 6, ...];
(1/3)*c = [0; 6, 1, 1, 2, 2, 2, 1, 3, 42, 5, 1, 2, 6, 2, 1, 7, ..., 3*(12*k + 2), 4*k + 1, 1, 2, 4*k + 2, 2, 1, 4*k + 3, ...]. (End)
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