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.

A272335 Decimal expansion of a function approximation constant which is the analog of Gibbs's constant 2*G/Pi (A036793) for de la Vallée-Poussin sums.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 26 2016

Keywords

Examples

			1.14272812693068128481021845956657111930110150452947023957171253...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham Constant, p. 248.

Links

Crossrefs

Cf. A036792, A036793, A243267, A243268, A245535.

Programs

  • Mathematica
    (2/Pi)(2 SinIntegral[4 Pi/3] - SinIntegral[2 Pi/3]) // N[#, 101]& // RealDigits // First

Formula

Equals (2/Pi)*Integral_{t=0..2*Pi/3} (cos(t) - cos(2*t))/t^2 dt.
Equals (2/Pi)*(2*Si(4*Pi/3) - Si(2*Pi/3)), where Si is the Sine integral function.

A243267 Decimal expansion of 1/2+G/Pi, the highest limiting crest of a square wave Fourier series, where G is the Gibbs-Wilbraham constant.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 02 2014

Keywords

Examples

			1.08948987223608363511601442291245487...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham constant, p. 249.

Links

Crossrefs

Cf. A036792, A036793, A243268.

Programs

  • Mathematica
    G = SinIntegral[Pi]; RealDigits[1/2 + G/Pi, 10, 99] // First

A243268 Decimal expansion of 1/2-G/Pi, the lowest limiting trough of a square wave Fourier series, where G is the Gibbs-Wilbraham constant. [negated].

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 02 2014

Keywords

Examples

			-0.08948987223608363511601442291245487...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham constant, p. 249.

Links

Crossrefs

Cf. A036792, A036793, essentially the same as A243267.

Programs

  • Mathematica
    G = SinIntegral[Pi]; RealDigits[1/2 - G/Pi, 10, 97] // First

A036791 Continued fraction for (2/Pi)*Integral_{x=0..Pi} sin(x)/x.

Original entry on oeis.org

1, 5, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 6, 1, 1, 1, 1, 1, 9, 24, 1, 3, 1, 7, 1, 5, 1, 2, 2, 3, 1, 2, 2, 1, 8, 11, 4, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 23, 1, 3, 3, 1, 6, 2, 9, 1, 3, 2, 17, 1, 5, 3, 1, 8, 1, 1, 1, 1, 1, 4, 1, 5, 1, 2, 1, 38, 1, 5, 5, 2, 6, 2, 73, 1, 1, 1, 194, 27, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Continued fraction expansion for Integrate[Binomial[1,x], {x,0,1}]. - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 13 2006
Integral(sin(x)/x dx) = x - x^3/(3*3!) + x^5/(5*5!) - x^7/(7*7!) + ... - Harry J. Smith, Apr 28 2009

Examples

			1.178979744472167270232028845... = 1 + 1/(5 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, Apr 28 2009

Links

Crossrefs

Cf. A036793 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[N[Integrate[Binomial[1, x], {x, 0, 1}], 120]] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 13 2006 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 21000); y=0; x=Pi; m=x; x2=x*x; n=1; nf=1; s=1; while (x!=y, y=x; n++; nf*=n; n++; nf*=n; m*=x2; s=-s; x+=s*m/(n*nf)); x*=2/Pi; x=contfrac(x); for (n=1, 20000, write("b036791.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 28 2009

Extensions

Offset changed by Andrew Howroyd, Aug 03 2024

A245535 Decimal expansion of the analog of the Gibbs-Wilbraham constant for L_1 trigonometric polynomial approximation.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jul 25 2014

Keywords

Examples

			x0 = 1.376991769203938865765266614301624670814900061506257246...
g(x0) = 0.0657838882664480990565512180874704669499564803216...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.1 Gibbs-Wilbraham Constant, p. 249.

Links

Crossrefs

Cf. A036792, A036793, A243267, A243268.

Programs

  • Mathematica
    digits = 101; g[x_] := (PolyGamma[x/2] - PolyGamma[(x+1)/2] + 1/x)*Sin[Pi*x]/Pi; x0 = x /. FindRoot[g'[x] == 0, {x, 3/2}, WorkingPrecision -> digits+5]; RealDigits[g[x0], 10, digits] // First

Formula

Maximum g(x0) of the function g(x) = (psi(x/2) - psi((x+1)/2) + 1/x)*sin(Pi*x)/Pi, for x >= 1, where psi is the polygamma function.
Showing 1-5 of 5 results.

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

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