%I A006752 M4593
%S A006752 9,1,5,9,6,5,5,9,4,1,7,7,2,1,9,0,1,5,0,5,4,6,0,3,5,1,4,9,3,2,3,8,4,1,1,
%T A006752 0,7,7,4,1,4,9,3,7,4,2,8,1,6,7,2,1,3,4,2,6,6,4,9,8,1,1,9,6,2,1,7,6,3,0,
%U A006752 1,9,7,7,6,2,5,4,7,6,9,4,7,9,3,5,6,5,1,2,9,2,6,1,1,5,1,0,6,2,4,8,5,7,4
%N A006752 Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 
               - ...
%C A006752 With the k-th appended term being 2*3*...*(2+k-2)*2^k*(2^k-1)*Bern(k) 
               / (2*k!*(J^(k+2-1))). Bern(k) is a Bernoulli number and J is a large 
               number of the form 4n + 1. This is from "An Atlas Of Functions" by 
               Spanier, J. and Oldham, K. B. 1987, equation 3:3:7. */ [From Harry 
               J. Smith (hjsmithh(AT)sbcglobal.net), May 07 2009]
%D A006752 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006752 G. J. Fee, ``Computation of Catalan's constant using Ramanujan's formula,
               '' in Proc. Internat. Symposium on Symbolic and Algebraic Computation 
               (ISSAC '90). 1990, pp. 157-160.
%D A006752 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and 
               its Applications, vol. 94, Cambridge University Press, pp. 53-59
%H A006752 Harry J. Smith, <a href="b006752.txt">Table of n, a(n) for n=0,...,20000</
               a>
%H A006752 Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/
               publist.html">Zeta function expansions of some classical constants</
               a>
%H A006752 T. Papanikolaou and G. Fee, <a href="http://www.gutenberg.org/etext/812">
               Catalan's Constant [Ramanujan's Formula] to 1,500,000 places</a> 
               [Gutenberg Project Etext]
%H A006752 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CatalansConstant.html">Link to a section of The World of Mathematics.</
               a>
%F A006752 Equals Integrate[ ArcTan[x]/x, {x,0,1}].
%F A006752 Equals Integrate[ 3 ArcTan[x(1-x)/(2-x)]/x, {x,0,1}] . - Posting to Number 
               Theory List by James McLaughlin, Sep 27 2007
%e A006752 0.915965...
%t A006752 nmax = 1000; First[ RealDigits[Catalan, 10, nmax] ] - Stuart Clary (clary(AT)uakron.edu), 
               Dec 17, 2008
%o A006752 (PARI) { digits=20000; default(realprecision, digits+80); s=1.0; n=5*digits; 
               j=4*n+1; si=-1.0; for (i=3, j-2, s+=si/i^2; si=-si; i++; ); s+=0.5/
               j^2; ttk=4.0; d=4.0*j^3; xk=2.0; xkp=xk; for (k=2, 100000000, term=(ttk-1)*ttk*xkp; 
               xk++; xkp*=xk; if (k>2, term*=xk; xk++; xkp*=xk; ); term*=bernreal(k)/
               d; sn=s+term; if (sn==s, break); s=sn; ttk*=4.0; d*=(k+1)*(k+2)*j^2; 
               k++; ); x=10*s; for (n=0, digits, d=floor(x); x=(x-d)*10; write("b006752.txt", 
               n, " ", d)); } /* Beta(2) = 1 - 1/3^2 + 1/5^2 - ... - 1/(J-2)^2 + 
               1/(2*J^2) + 2*Bern(0)/(2*J^3) - 2*3*4*Bern(2)/J^5 + ... ,
%Y A006752 Cf. A014538, A104338, A014538, A153069, A153070, A054543, A118323.
%Y A006752 Sequence in context: A143296 A021526 A019791 this_sequence A164802 A090656 
               A058284
%Y A006752 Adjacent sequences: A006749 A006750 A006751 this_sequence A006753 A006754 
               A006755
%K A006752 nonn,cons,easy
%O A006752 0,1
%A A006752 N. J. A. Sloane (njas(AT)research.att.com).
%E A006752 More terms from Larry Reeves (larryr(AT)acm.org), May 28 2002
%E A006752 Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 19 2009