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 1-4 of 4 results.

A111129 Decimal expansion of the continued fraction 1+1/(1+2/(1+3/(1+4/(1+5/(1+...))))).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, based on correspondence from Tom Raes (tommy1729(AT)hotmail.com) and Steven Finch, Sep 22 2005

Keywords

Examples

			1.52513527616098120908909053639057871330711636492060333554631394242...

References

  • B. C. Berndt, Y.-S. Choi and S.-Y. Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society, Continued Fractions: From Analytic Number Theory to Constructive Approximation, ed. B. C. Berndt and F. Gesztesy, Amer. Math. Soc., 1999, pp. 15-56.
  • S. R. Finch, "Mathematical Constants", Cambridge, pp. 423-428.
  • H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, 1948, pp. 356-358, 367

Links

Crossrefs

Cf. A099287, A108088, A111188.
Cf. A225435, A225436 (numerators and denominators of convergents to c.f.).

Programs

  • Mathematica
    RealDigits[1/(Sqrt[Pi*E/2]*Erfc[1/Sqrt[2]]), 10, 111][[1]]
  • PARI
    1/(sqrt(Pi*exp(1)/2)*erfc(1/sqrt(2))) \\ G. C. Greubel, Jan 24 2017

Formula

Equals the reciprocal of sqrt(pi*e/2)*erfc(1/sqrt(2)), where erfc is the complementary error function.

Extensions

More terms from Robert G. Wilson v and Hans Havermann, Oct 17 2005
Definition corrected by Steven Finch, Feb 05 2009

A059444 Decimal expansion of square root of (Pi * e / 2).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Feb 01 2001

Keywords

Comments

Appears as constant factor in Proposition 1.12, p. 5, of Feige et al. (2007). - Jonathan Vos Post, Jun 18 2007

Examples

			2.066365677...

References

  • C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, Oxford and NY, 2001, page 68.

Links

Crossrefs

Cf. A059445, A060196, A108088.

Programs

  • Mathematica
    RealDigits[N[Sqrt[ \[Pi]*\[ExponentialE]/2], 100]][[1]]
RealDigits[Sqrt[(Pi*E)/2],10,120][[1]] (* Harvey P. Dale, Nov 27 2024 *)
  • PARI
    { default(realprecision, 20080); x=sqrt(Pi*exp(1)/2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b059444.txt", n, " ", d)); } \\ Harry J. Smith, Jun 27 2009

Formula

Sqrt(Pi*e/2) = A + B with A = 1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) + 1/(1*3*5*7*9) + ... = 1.410686134... (see A060196) and B = 1/(1 + 1/(1 + 2/(1 + 3/(1 + 4/(1 + 5/(1 + ...)))))) = 0.65567954241... (see A108088) - (S. Ramanujan)
Equals (sqrt(2)*exp(1/4)*(sum(n>=0, n!/(2*n)! ) - 1))/erf(1/2). - Jean-François Alcover, Mar 22 2013

Extensions

Edited by Daniel Forgues, Apr 14 2011

A289491 a(n) = denominator of 1/(1 + 1/(1 + 2/(1 + ... (1 + n)))).

Original entry on oeis.org

2, 4, 5, 13, 19, 58, 191, 131, 1187, 2231, 17519, 71063, 29881, 323423, 2887921, 13237457, 2397389, 15030317, 742458253, 3748521653, 9670072483, 25451905333, 10932619111, 78684575461, 4163946939067, 11799518538967, 136025604432743, 159359728522979
Offset: 1

Views

Author

Seiichi Manyama, Sep 02 2017

Keywords

Examples

			1/2, 3/4, 3/5, 9/13, 12/19, 39/58, 123/191, 87/131, 771/1187, 1473/2231, 11427/17519, 46779/71063, 19533/29881, ... = A225436/A289491 -> A108088.
A225436(1)/a(1) = 1/2  = 1/(1 + 1)                         =  1/2,
A225436(2)/a(2) = 3/4  = 1/(1 + 1/(1 + 2))                 =  3/4,
A225436(3)/a(3) = 3/5  = 1/(1 + 1/(1 + 2/(1 + 3)))         =  6/10,
A225436(4)/a(4) = 9/13 = 1/(1 + 1/(1 + 2/(1 + 3/(1 + 4)))) = 18/26.

Links

Crossrefs

Cf. A000085, A000932, A108088, A225435, A225436 (numerators).

Programs

  • Maple
    p:= (i, n)-> `if`(i=n, (1+n), 1+i/p(i+1,n)):
a:= n-> denom(1/p(1,n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 02 2017

Formula

a(n) = A225435(n) + A225436(n).
A225436(n)/a(n) = 1/(1 + 1/(1 + 2/(1 + ... (1 + n)))) = A000932(n)/A000085(n+1).

A257526 Decimal expansion of e*Pi*erfc(1).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 28 2015

Keywords

Examples

			1.343293421646735170437123594410589778322829567130036872051955564553...

Links

Crossrefs

Cf. A099287, A108088.

Programs

  • Mathematica
    RealDigits[E*Pi*Erfc[1], 10, 103] // First
  • PARI
    exp(1)*Pi*erfc(1) \\ Charles R Greathouse IV, Apr 18 2016

Formula

Equals Integral_{-infinity..infinity} exp(-x^2)/(1+x^2) dx.
Also equals J(0) where J(c) = Integral_{-infinity..infinity} exp(-(x-c)^2)/(1+x^2) dx = (1/2)*Pi*e*(erfc[1-c*i]*e^(-2*c*i) + erfc[1+c*i]*e^(2*c*i)), where the integrand comes from a shifted normal PDF times a Cauchy PDF.
Equals 2 * Integral_{x=0..Pi/2} exp(-tan(x)^2) dx. - Amiram Eldar, Aug 07 2020
Showing 1-4 of 4 results.

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

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