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A006752
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Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...
(Formerly M4593)
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44
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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, 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, 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
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OFFSET
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0,1
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COMMENTS
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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. */ [Harry J. Smith, May 07 2009]
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REFERENCES
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Victor Adamchik: 33 representations for Catalan's constant, http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
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.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 53-59
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0..20000
Greg Fee, Project Gutenberg, Catalan's Constant to 300000 digits
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
F. M. S. Lima, A rapidly converging Ramanujan-type series for Catalan's constant, arXiv:1207.3139v1 [math.NT], July 13, 2012.
A. Lupas, Formulae for some classical constants, in Proceedings of ROGER-2000, 2000.
T. Papanikolaou and G. Fee, Catalan's Constant [Ramanujan's Formula] to 1,500,000 places [Gutenberg Project Etext]
_Simon Plouffe_, Generalized expansions of real numbers, 2006.
Eric Weisstein's World of Mathematics, Catalan's Constant.
Wikipedia, Catalan's constant
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FORMULA
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c = integrate(x=0..1, arctan(x)/x ).
c = integrate(x=0..1, 3*arctan(x*(1-x)/(2-x))/x ). - Posting to Number Theory List by James McLaughlin, Sep 27 2007
c = (zeta(2,1/4)- zeta(2,3/4))/16 - [Gerry Martens, May 27 2011]
c = 1/2*sum(n>=0, (-1)^n * ((3*n+2)*8^n) / ((2*n+1)^3*C(2*n,n)^3) ) (from the Lima 2012 reference).
c = -1/64*sum(n>=1, (-1)^n * (2^(8*n) * (40*n^2-24*n+3)) / (n^3 * (2*n-1) * C(2*n,n) * C(4*n,2*n)^2) ) (from the Lupas 2000 reference).
c = phi(-1, 2, 1/2)/4, where phi is Lerch transcendent. [Jean-François Alcover, Mar 28 2013]
c = 1/2*integrate(x=0..Pi/2, log(cot(x)+csc(x)) ). [Jean-François Alcover, Apr 11 2013]
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EXAMPLE
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0.9159655941772190150546035149323841107741493742816721342664981196217630197762\
547694793565129261151...
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MAPLE
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evalf(Catalan) ; # R. J. Mathar, Apr 09 2013
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MATHEMATICA
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nmax = 1000; First[ RealDigits[Catalan, 10, nmax] ] (* Stuart Clary, Dec 17 2008 *)
Integrate[ArcTan[x]/x, {x, 0, 1}] (* N. J. A. Sloane, May 03 2013*)
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PROG
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(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 + ... ,
(PARI) default(realprecision, 1000+2); /* 1000 terms */
s=sumalt(n=0, (-1)^n/(2*n+1)^2);
v=Vec(Str(s)); /* == ["0", ".", "9", "1", "5", "9", "6", ...*/
vector(#v-2, n, eval(v[n+2]))
/* Joerg Arndt, Aug 25 2011 */
(PARI) Catalan \\ Charles R Greathouse IV, Nov 20 2011
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CROSSREFS
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Cf. A014538, A104338, A014538, A153069, A153070, A054543, A118323.
Sequence in context: A205326 A021526 A019791 * A164802 A201888 A185825
Adjacent sequences: A006749 A006750 A006751 * A006753 A006754 A006755
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KEYWORD
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nonn,cons,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), May 28 2002
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STATUS
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approved
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