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.

A002161 Decimal expansion of square root of Pi.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Also Gamma(1/2). - Franklin T. Adams-Watters, Apr 07 2006
The integral of the Gaussian function exp(-x^2) over the real line. - Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008
Also equals the average distance between two points in two dimensions where coordinates are independent normally distributed random variables with mean 0 and variance 1. - Jean-François Alcover, Oct 31 2014, after Steven Finch
Also diameter of a sphere whose surface area equals Pi^2. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018
Convergents of continued fractions: 7/4, 16/9, 23/13, 39/22, 257/145, 296/167, 8545/4821, ... - R. J. Mathar, Jan 29 2025

Examples

			1.7724538509055160272981674833411451827975494561223871282138...

References

  • George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 190.
  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
  • W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, page 413.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 40.

Links

Crossrefs

Cf. A000796, A002388, A073006, A195907, A203145.
Cf. decimal expansions of Gamma(1/k): A073005 (k=3), A068466 (k=4), A175380 (k=5), A175379 (k=6), A220086 (k=7), A203142 (k=8).

Programs

  • Magma
    R:= RealField(100); Sqrt(Pi(R));  // G. C. Greubel, Mar 10 2018
  • Maple
    evalf(sqrt(Pi),120); # Muniru A Asiru, Nov 11 2018
  • Mathematica
    RealDigits[N[Sqrt[Pi], 120]][[1]] (* Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008 *)
  • PARI
    default(realprecision, 20080); x=sqrt(Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002161.txt", n, " ", d)); \\ Harry J. Smith, May 01 2009

Formula

Equals (1/2) * Sum_{n>=0} ((-1)^n * (4*n+1) * (1/8)^(n+1) * (2^(n+1))^3 * Gamma(n+1/2)^3 / Gamma(n+1)^3). - Alexander R. Povolotsky, Mar 25 2013
Equals Integral_{x=0..1} 1/sqrt(-log(x)) dx. - Jean-François Alcover, Apr 29 2013
Equals Sum_{k>=0} (k+1/2)!/(k+2)!. - Amiram Eldar, Jun 19 2023
Equals Integral_{x=0..oo} exp(-x)/sqrt(x) dx. - Michal Paulovic, Sep 24 2023
Equals Integral_{x=0..oo} 4/(exp(x^2)*(2*x^2 + 1)^2) dx. - Kritsada Moomuang, Jun 05 2025

Extensions

More terms from Franklin T. Adams-Watters, Apr 07 2006

A218792 Decimal expansion of Sum_{n = -oo..oo} e^(-2*n^2).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Nov 05 2012

Keywords

Examples

			1.2713415221890152252223825787909356249768649877176...
For comparison, sqrt(Pi/2) = 1.2533141373155002512078826424055226265034933703050...

Links

Crossrefs

Cf. A195907, A069998, A167009, A167010, A229052.

Programs

  • Mathematica
    RealDigits[Sum[E^(-2*k^2), {k,-Infinity,Infinity}], 10, 200][[1]]
RealDigits[EllipticTheta[3,0,1/E^2],10,200][[1]] (* Vaclav Kotesovec, Sep 22 2013 *)
  • PARI
    1 + 2*suminf(n=1, exp(-2*n^2)) \\ Charles R Greathouse IV, Jun 06 2016
  • PARI
    (eta(2*I/Pi))^5 / (eta(I/Pi)^2 * eta(4*I/Pi)^2) \\ Jianing Song, Oct 13 2021

Formula

Equals Jacobi theta_{3}(0,exp(-2)). - G. C. Greubel, Feb 01 2017
Equals eta(2*i/Pi)^5 / (eta(i/Pi)*eta(4*i/Pi))^2, where eta(t) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + ... is the Dedekind eta function without the q^(1/24) factor in powers of q = exp(2*Pi*i*t) (Cf. A000122). - Jianing Song, Oct 14 2021

A348333 Decimal expansion of Sum_{n=-oo..oo} 1/(n^2)!.

Original entry on oeis.org

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

Views

Author

Jianing Song, Oct 13 2021

Keywords

Comments

This constant is irrational. The proof is similar to the one that e is irrational. Is this constant transcendental?

Examples

			3.083338844797273...

Crossrefs

Cf. A195907, A001113 (e = Sum_{n>=0} 1/n!).

Programs

  • Mathematica
    RealDigits[N[1 + 2 *Sum[1/(n^2)!, {n, 1, Infinity}], 90]][[1]] (* Amiram Eldar, Oct 14 2021 *)
  • PARI
    1 + 2*suminf(n=1, 1/(n^2)!)
Showing 1-3 of 3 results.

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

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