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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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108
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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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Bailey, David H., Jonathan M. Borwein, Andrew Mattingly, and Glenn Wightwick. "The Computation of Previously Inaccessible Digits of π^2 and Catalan’s Constant," Notices AMS, 60 (No. 7 2013), 844-854.
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
Victor Adamchik, 33 representations for Catalan's constant
Greg Fee, Project Gutenberg, Catalan's Constant to 300000 digits
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.
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
Werner Hürlimann, Exact and Asymptotic Evaluation of the Number of Distinct Primitive Cuboids, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.5.
Yasuyuki Kachi and Pavlos Tzermias, Infinite products involving zeta(3) and Catalan's constant, Journal of Integer Sequences, Vol. 15 (2012), #12.9.4.
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.
Xiaohan Wang, The Barnes G-function and the Catalan Constant, Kyushu Journal of Mathematics, Vol. 67 (2013) No. 1 p. 105-116.
Eric Weisstein's World of Mathematics, Catalan's Constant
Eric Weisstein's World of Mathematics, Catalan's Constant Digits
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 [see the Adamchik link]
c = - integrate(x=0..1,(log x)/(1+x^2)) = integrate(x=1..oo,(log x)/(1+x^2)). - Clark Kimberling, Nov 04 2016
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EXAMPLE
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0.91596559417721901505460351493238411077414937428167213426649811962176301977...
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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 * A271856 A164802 A201888
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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