A019727 Decimal expansion of sqrt(2*Pi).

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

Offset

1,1

Comments

Pickover says that the expression: lim_{n->oo} e^n(n!) / (n^n * sqrt(n)) = sqrt(2*Pi) is beautiful because it connects Pi, e, radicals, factorials and infinite limits. - Jason Earls, Mar 16 2001

Appears in the formula of the normal distribution. - Johannes W. Meijer, Feb 23 2013

sqrt(2*Pi)*sqrt(n) is the expected height of a labeled random tree of order n (see Rényi, Szekeres, 1967, formula (4.6)). - Hugo Pfoertner, May 18 2023

The constant in the formula known as "Stirling's approximation" (or "Stirling's formula"). It is sometimes called Stirling constant. The formula without the exact value of the constant was discovered by the French mathematician Abraham de Moivre (1667-1754), and was published in his book (1730). The exact value of the constant was found by the Scottish mathematician James Stirling (1692-1770) and was published in his book "Methodus differentialis" (1730). - Amiram Eldar, Jul 08 2023

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 307.

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 62-64.

Abraham de Moivre, Miscellanea Analytica de Seriebus et Quadraturis, London, England: J. Tonson & J. Watts, 1730, pp. 96-106.

K. Kimoto, N. Kurokawa, C. Sonoki, and M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.

Clifford A. Pickover, Zentralblatt review of Wonders of Numbers, Adventures in Mathematics, Mind and Meaning.

A. Rényi and G. Szekeres, On the height of trees, Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 497-507.

James Stirling, Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum, London, 1730. See Propositio XXVIII, pp. 135-139.

Eric Weisstein's World of Mathematics, Normal Distribution.

Wikipedia, Stirling's approximation.

Index entries for transcendental numbers.

Formula

Equals lim_{n->oo} e^n*(n!)/n^n*sqrt(n).

Also equals Integral_{x >= 0} W(1/x^2) where W is the Lambert function, which is also known as ProductLog. - Jean-François Alcover, May 27 2013

Also equals the generalized Glaisher-Kinkelin constant A_0, see the Finch reference. - Jean-François Alcover, Dec 23 2014

Equals exp(-zeta'(0)). See Kimoto et al. - Michel Marcus, Jun 27 2019

Example

2.506628274631000502415765284811045253006986740609938316629923576342293....

Mathematica

RealDigits[Sqrt[2Pi], 10, 120][[1]] (* Harvey P. Dale, Dec 12 2012 *)

Prog

(PARI) default(realprecision, 20080); x=sqrt(2*Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019727.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009
(Maxima) fpprec: 100$ ev(bfloat(sqrt(2*%pi))); /* Martin Ettl, Oct 11 2012 */
(Magma) R:= RealField(100); Sqrt(2*Pi(R)); // G. C. Greubel, Mar 08 2018

Crossrefs

Cf. A058293 (continued fraction), A231863 (inverse), A000796 (Pi).

Sequence in context: A021403 A299623 A290796 * A011184 A157214 A066033

Adjacent sequences: A019724 A019725 A019726 * A019728 A019729 A019730

Keyword

nonn,cons

Author

N. J. A. Sloane

Status

approved