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 A002388 Decimal expansion of Pi^2. (Formerly M4596 N1961) 31

%I M4596 N1961

%S 9,8,6,9,6,0,4,4,0,1,0,8,9,3,5,8,6,1,8,8,3,4,4,9,0,9,9,9,8,7,6,1,5,1,

%T 1,3,5,3,1,3,6,9,9,4,0,7,2,4,0,7,9,0,6,2,6,4,1,3,3,4,9,3,7,6,2,2,0,0,

%U 4,4,8,2,2,4,1,9,2,0,5,2,4,3,0,0,1,7,7,3,4,0,3,7,1,8,5,5,2,2,3,1,8,2,4,0,2

%N Decimal expansion of Pi^2.

%C Also equals the volume of revolution of the sine or cosine curve for one full period,Integral_{0,2Pi} Sin(x)^2 dx. - Robert G. Wilson v Dec 15 2005. - Robert G. Wilson v, Dec 15 2005

%C Also equals 32*Integral_{0, 1} ArcTan(x)/(1+x^2) dx [_Jean-François Alcover_, Mar 25 2013]

%D Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.

%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="/A002388/b002388.txt">Table of n, a(n) for n=1,...,20000</a>

%H D. H. Bailey and J. M. Borwein, <a href="http://eprints.cecm.sfu.ca/archive/00000269/">Experimental Mathematics: Examples, Methods and Implications</a>

%H N. D. Elkies, <a href="http://www.math.harvard.edu/~elkies/Misc/pi10.pdf">Why is (pi)^2 so close to 10?</a>

%H _Simon Plouffe_, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap75.html">Pi^2 to 10000 digits</a>

%H _Simon Plouffe_, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/pipi.txt">Pi^2 to 10000 digits</a>

%H <a href="/index/Ph#Pi314">Index entries for sequences related to the number Pi</a>

%F Pi^2 = 11/2 + 16 * sum(k>=2, (1+k-k^3)/(1-k^2)^3 ) [From Alexander R. Povolotsky, May 04 2009]

%F Pi^2 = 3*(sum(n>=1, (2*n+1)^2/(sum(k=1..n, k^3 )) )/4 - 1) [From Alexander R. Povolotsky, Jan 14 2011]

%F Pi^2 = 3/2*(sum(n>=1, (7*n^2+2*n-2)/(2*n^2-1)/(n+1)^5 ) -zeta(3) -3*zeta(5)+22-7*polygamma(0,1-1/sqrt(2)) +5*sqrt(2)*polygamma(0,1-1/sqrt(2)) -7*polygamma(0,1+1/sqrt(2)) -5*sqrt(2)*polygamma(0,1+1/sqrt(2)) -14*EulerGamma) [From Alexander R. Povolotsky, Aug 13 2011]

%e 9.869604401089358618834490999876151135313699407240790626413349376220044... [From Harry J. Smith, May 31 2009]

%p Pi^2 = 11/2 + 16 * sum(k>=2, (1+k-k^3)/(1-k^2)^3 ) [From Alexander R. Povolotsky, May 04 2009]

%t RealDigits[Pi^2, 10, 111][[1]] (* Robert G. Wilson v *)

%o (PARI) { default(realprecision, 20080); x=Pi^2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002388.txt", n, " ", d)); } [From _Harry J. Smith_, May 31 2009]

%Y Cf. A102753.

%Y Cf. A058284 Continued fraction. [From _Harry J. Smith_, May 31 2009]

%K nonn,cons

%O 1,1

%A _N. J. A. Sloane_.

%E More terms from _Robert G. Wilson v_, Dec 15 2005

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