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A019727 Decimal expansion of sqrt(2*Pi). 36

%I

%S 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,

%T 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,

%U 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

%N Decimal expansion of sqrt(2*Pi).

%C 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

%C Appears in the formula of the normal distribution. - _Johannes W. Meijer_, Feb 23 2013

%D 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.

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

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

%H Harry J. Smith, <a href="/A019727/b019727.txt">Table of n, a(n) for n = 1..20000</a>

%H K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, <a href="http://www.math.u-ryukyu.ac.jp/~kimoto/pdf/kksw2.pdf">Some examples of generalized zeta regularized products</a>, Kodai Math. J. 27 (2004), 321-335.

%H C. A. Pickover, <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&amp;format=complete">Zentralblatt review of Wonders of Numbers, Adventures in Mathematics, Mind and Meaning</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/NormalDistribution.html">MathWorld: Normal Distribution</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

%F Also equals Integral_{x >= 0} W(1/x^2) where W is the Lambert function, which is also known as ProductLog. - _Jean-Fran├žois Alcover_, May 27 2013

%F Also equals the generalized Glaisher-Kinkelin constant A_0, see the Finch reference - _Jean-Fran├žois Alcover_, Dec 23 2014

%F Equals exp(-zeta'(0)). See Kimoto et al. - _Michel Marcus_, Jun 27 2019

%e 2.506628274631000502415765284811045253006986740609938316629923576342293....

%t RealDigits[Sqrt[2Pi],10,120][[1]] (* _Harvey P. Dale_, Dec 12 2012 *)

%o (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

%o (Maxima) fpprec: 100$ ev(bfloat(sqrt(2*%pi))); /* _Martin Ettl_, Oct 11 2012 */

%o (MAGMA) R:= RealField(100); Sqrt(2*Pi(R)); // _G. C. Greubel_, Mar 08 2018

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

%K nonn,cons,changed

%O 1,1

%A _N. J. A. Sloane_

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Last modified July 13 13:48 EDT 2020. Contains 335688 sequences. (Running on oeis4.)