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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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199
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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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Usually denoted by G.
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. See equation 3:3:7 in Spanier and Oldham. - Harry J. Smith, May 07 2009
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 57, 554.
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).
J. Spanier, and K. B. Oldham, An Atlas of Functions, 1987, equation 3:3:7.
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 0..20000
Milton Abramowitz and Irene A. Stegun, editors, Catalan's constant, Handbook of Mathematical Functions, December 1972, p. 807, 23.2.21 for n=2.
Victor Adamchik, 33 representations for Catalan's constant.
David H. Bailey, Jonathan M. Borwein, Andrew Mattingly, and Glenn Wightwick, The Computation of Previously Inaccessible Digits of Pi^2 and Catalan’s Constant, Notices AMS, 60 (No. 7 2013), 844-854.
Peter Bala, New series for old functions
David M. Bradley, Representations of Catalan's constant.
Sarth Chavan and Christophe Vignat, A Triple Integral representation of Catalan's constant, arXiv:2105.11771 [math.NT], 2021.
Greg Fee, Catalan's Constant to 300000 digits, Project Gutemberg.
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], Jul 13 2012.
A. Lupas, Formulae for some classical constants, in Proceedings of ROGER-2000, 2000. [Local copy]
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
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Trigamma Function
Wikipedia, Catalan's constant
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FORMULA
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G = Integral_{x=0..1} arctan(x)/x dx.
G = Integral_{x=0..1} 3*arctan(x*(1-x)/(2-x))/x dx. - Posting to Number Theory List by James McLaughlin, Sep 27 2007
G = (zeta(2,1/4)- zeta(2,3/4))/16. - Gerry Martens, May 27 2011 [With the Hurwitz Zeta function zeta.]
G = (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).
G = (-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).
G = phi(-1, 2, 1/2)/4, where phi is Lerch transcendent. - Jean-François Alcover, Mar 28 2013
G = (1/2)*Integral_{x=0..Pi/2} log(cot(x)+csc(x)) dx. - Jean-François Alcover, Apr 11 2013 [see the Adamchik link]
G = -Integral_{x=0..1} (log x)/(1+x^2) dx = Integral_{x>=1} (log x)/(1+x^2) dx. - Clark Kimberling, Nov 04 2016
G = (Zeta(2, 1/4) - Pi^2)/8 = (Psi(1, 1/4) - Pi^2)/8, with the Hurwitz Zeta function and the trigamma function Psi(1, z). For the partial sums of the series given in the name see A294970/A294971. - Wolfdieter Lang, Nov 15 2017
Equals Im(Li_{2}(i)). - Peter Luschny, Oct 04 2019
Equals -Integral_{x=0..Pi/4} log(tan(x)) dx. - Amiram Eldar, Jun 29 2020
Equals (1/2)*Integral_{x=0..1} K(x) dx = -1/2 + Integral_{x=0..1} E(x) dx, where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively, as a functions of the elliptic modulus k. - Gleb Koloskov, Jun 25 2021
From Peter Bala, Dec 08 2021: (Start)
G = 1/2 + 4*Sum_{n >= 1} (-1)^(n+1)*n/(4*n^2 - 1)^2 = -13/18 + (2^7)*3*Sum_{n >= 1} (-1)^(n+1)*n/((4*n^2 - 1)^2*(4*n^2 - 9)^2) = -3983/1350 + (2^15)*3*5*Sum_{n >= 1} (-1)^(n+1)*n/((4*n^2 - 1)^2*(4*n^2 - 9)^2*(4*n^2 - 25)^2).
G = 3/2 - 16*Sum_{n >= 1} (-1)^(n+1)*n/(4*n^2 - 1)^3 = 401/6 - (2^13)*(3^3)*Sum_{n >= 1} (-1)^n*n/((4*n^2 - 1)^3*(4*n^2 - 9)^3) = 5255281/1350 - (2^25)*(3^3)*(5^3)*Sum_{n >= 1} (-1)^(n+1)*n/((4*n^2 - 1)^3*(4*n^2 - 9)^3*(4*n^2 - 25)^3). (End)
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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 *)
N[Im[PolyLog[2, I]], 100] (* Peter Luschny, Oct 04 2019 *)
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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
(PARI) (zetahurwitz(2, 1/4)-Pi^2)/8 \\ Charles R Greathouse IV, Jan 30 2018
(Magma) R:= RealField(100); Catalan(R); // G. C. Greubel, Aug 21 2018
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CROSSREFS
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Cf. A014538, A104338, A153069, A153070, A054543, A118323, A294970/A294971.
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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