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%I M4332 N1814 #78 Sep 24 2023 09:21:15
%S 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,
%T 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,
%U 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
%N Decimal expansion of square root of Pi.
%C Also Gamma(1/2). - _Franklin T. Adams-Watters_, Apr 07 2006
%C The integral of the Gaussian function exp(-x^2) over the real line. - Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008
%C 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_
%C 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
%D George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 190.
%D W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Harry J. Smith, <a href="/A002161/b002161.txt">Table of n, a(n) for n = 1..20000</a>
%H Donald Knuth, <a href="http://www.youtube.com/watch?v=cI6tt9QfRdo">Why pi?</a>, Christmas Tree lecture, Dec 06 2010 (video).
%H <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F 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
%F Equals Integral_{x=0..1} 1/sqrt(-log(x)) dx. - _Jean-François Alcover_, Apr 29 2013
%F Equals Sum_{k>=0} (k+1/2)!/(k+2)!. - _Amiram Eldar_, Jun 19 2023
%F Equals Integral_{x=0..oo} exp(-x)/sqrt(x) dx. - _Michal Paulovic_, Sep 24 2023
%e 1.7724538509055160272981674833411451827975494561223871282138...
%p evalf(sqrt(Pi),120); # _Muniru A Asiru_, Nov 11 2018
%t RealDigits[N[Sqrt[Pi], 120]][[1]] (* Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008 *)
%o (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
%o (Magma) R:= RealField(100); Sqrt(Pi(R)); // _G. C. Greubel_, Mar 10 2018
%Y Cf. A000796, A002388, A073006, A195907, A203145.
%Y 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).
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Franklin T. Adams-Watters_, Apr 07 2006
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