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A019727 Decimal expansion of sqrt(2*Pi). 11
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Pickover says that the expression: lim(n -> infinity) e^n(n!) / (n^n * sqrt(n)) = sqrt(2*Pi) is beautiful because it connects Pi, e, radicals, factorials and infinite limits. - Jason Earls (zevi_35711(AT)yahoo.com), Mar 16 2001

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

REFERENCES

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

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

LINKS

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

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

Eric W. Weisstein, MathWorld: Normal Distribution

FORMULA

Equal to lim(n -> infinity) e^n*(n!)/n^n*sqrt(n).

Also equals integral_{x = 0..infinity} W(1/x^2) where W is the Lambert function, which is also known as ProductLog. [Jean-Fran├žois Alcover, May 27 2013]

EXAMPLE

2.506628274631000502415765284811045253006986740609938316629923576342293... [Harry J. Smith, May 31 2009]

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 */

CROSSREFS

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

Sequence in context: A242969 A090625 A021403 * A011184 A157214 A066033

Adjacent sequences:  A019724 A019725 A019726 * A019728 A019729 A019730

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified September 15 23:11 EDT 2014. Contains 246792 sequences.