

A019727


Decimal expansion of sqrt(2*Pi).


36



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
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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, 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. 6264.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 GlaisherKinkelin constant, p. 137.
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
K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321335.
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} W(1/x^2) where W is the Lambert function, which is also known as ProductLog.  JeanFrançois Alcover, May 27 2013
Also equals the generalized GlaisherKinkelin constant A_0, see the Finch reference  JeanFrançois Alcover, Dec 23 2014
Equals exp(zeta'(0)). See Kimoto et al.  Michel Marcus, Jun 27 2019


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=(xd)*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



