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A006752 Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...
(Formerly M4593)
228

%I M4593 #223 Feb 21 2024 15:08:21

%S 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,

%T 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,

%U 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

%N Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...

%C Usually denoted by G.

%C 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

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 57, 554.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 53-59.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Jerome Spanier and Keith B. Oldham, An Atlas of Functions, 1987, equation 3:3:7.

%H Harry J. Smith, <a href="/A006752/b006752.txt">Table of n, a(n) for n = 0..20000</a>

%H Milton Abramowitz and Irene A. Stegun, editors, <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=807&amp;Submit=Go">Catalan's constant</a>, Handbook of Mathematical Functions, December 1972, p. 807, 23.2.21 for n=2.

%H Victor Adamchik, <a href="https://library.wolfram.com/infocenter/Demos/109/">33 representations for Catalan's constant</a>.

%H David H. Bailey, Jonathan M. Borwein, Andrew Mattingly, and Glenn Wightwick, <a href="http://www.ams.org/notices/201307/rnoti-p844.pdf">The Computation of Previously Inaccessible Digits of Pi^2 and Catalan's Constant</a>, Notices AMS, 60 (No. 7 2013), 844-854.

%H Peter Bala, <a href="/A002117/a002117.pdf">New series for old functions</a>.

%H David M. Bradley, <a href="https://www.researchgate.net/publication/2325473_Representations_of_Catalan%27s_Constant">Representations of Catalan's constant</a>, 2001.

%H Sarth Chavan and Christophe Vignat, <a href="https://arxiv.org/abs/2105.11771">A Triple Integral representation of Catalan's constant</a>, arXiv:2105.11771 [math.NT], 2021.

%H Greg Fee, <a href="https://www.gutenberg.org/ebooks/682">Catalan's Constant to 300000 digits</a>, Project Gutenberg, 1996.

%H G. J. Fee, <a href="http://dx.doi.org/10.1145/96877.96917">Computation of Catalan's constant using Ramanujan's formula</a>, in Proc. Internat. Symposium on Symbolic and Algebraic Computation (ISSAC '90). 1990, pp. 157-160.

%H Philippe Flajolet and Ilan Vardi, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">Zeta function expansions of some classical constants</a>.

%H Werner Hürlimann, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Huerlimann/huerli6.html">Exact and Asymptotic Evaluation of the Number of Distinct Primitive Cuboids</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.5.

%H Yasuyuki Kachi and Pavlos Tzermias, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Tzermias/tzermias2.html">Infinite products involving zeta(3) and Catalan's constant</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.9.4.

%H F. M. S. Lima, <a href="http://arxiv.org/abs/1207.3139">A rapidly converging Ramanujan-type series for Catalan's constant</a>, arXiv:1207.3139v1 [math.NT], Jul 13 2012.

%H A. Lupas, <a href="/A006752/a006752.pdf">Formulae for some classical constants</a>, in Proceedings of ROGER-2000, 2000. [Local copy]

%H David Naccache and Ofer Yifrach-Stav, <a href="https://arxiv.org/abs/2210.15669">On Catalan Constant Continued Fractions</a>, arXiv:2210.15669 [cs.SC], 2022.

%H 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>, Project Gutenberg, 1997.

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/gendev/915965.html">Generalized expansions of real numbers</a>, 2006.

%H Xiaohan Wang, <a href="http://doi.org/10.2206/kyushujm.67.105">The Barnes G-function and the Catalan Constant</a>, Kyushu Journal of Mathematics, Vol. 67 (2013) No. 1, pp. 105-116.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalansConstant.html">Catalan's Constant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalansConstantDigits.html">Catalan's Constant Digits</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrigammaFunction.html">Trigamma Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Catalan%27s_constant">Catalan's constant</a>.

%F G = Integral_{x=0..1} arctan(x)/x dx.

%F G = Integral_{x=0..1} 3*arctan(x*(1-x)/(2-x))/x dx. - Posting to Number Theory List by _James Mc Laughlin_, Sep 27 2007

%F G = (zeta(2,1/4)- zeta(2,3/4))/16. - _Gerry Martens_, May 27 2011 [With the Hurwitz zeta function zeta.]

%F 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).

%F 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).

%F G = phi(-1, 2, 1/2)/4, where phi is Lerch transcendent. - _Jean-François Alcover_, Mar 28 2013

%F G = (1/2)*Integral_{x=0..Pi/2} log(cot(x)+csc(x)) dx. - _Jean-François Alcover_, Apr 11 2013 [see the Adamchik link]

%F 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

%F 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

%F Equals Im(Li_{2}(i)). - _Peter Luschny_, Oct 04 2019

%F Equals -Integral_{x=0..Pi/4} log(tan(x)) dx. - _Amiram Eldar_, Jun 29 2020

%F 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

%F From _Peter Bala_, Dec 08 2021: (Start)

%F 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).

%F 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)

%F From _Amiram Eldar_, Jan 07 2024: (Start)

%F Equals beta(2), where beta is the Dirichlet beta function.

%F Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^2)^(-1). (End)

%e 0.91596559417721901505460351493238411077414937428167213426649811962176301977...

%p evalf(Catalan) ; # _R. J. Mathar_, Apr 09 2013

%t nmax = 1000; First[RealDigits[Catalan, 10, nmax]] (* _Stuart Clary_, Dec 17 2008 *)

%t Integrate[ArcTan[x]/x, {x, 0, 1}] (* _N. J. A. Sloane_, May 03 2013 *)

%t N[Im[PolyLog[2, I]], 100] (* _Peter Luschny_, Oct 04 2019 *)

%o (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 + ... ,

%o (PARI) default(realprecision,1000+2); /* 1000 terms */

%o s=sumalt(n=0,(-1)^n/(2*n+1)^2);

%o v=Vec(Str(s)); /* == ["0", ".", "9", "1", "5", "9", "6", ...*/

%o vector(#v-2,n,eval(v[n+2]))

%o /* _Joerg Arndt_, Aug 25 2011 */

%o (PARI) Catalan \\ _Charles R Greathouse IV_, Nov 20 2011

%o (PARI) (zetahurwitz(2,1/4)-Pi^2)/8 \\ _Charles R Greathouse IV_, Jan 30 2018

%o (Magma) R:= RealField(100); Catalan(R); // _G. C. Greubel_, Aug 21 2018

%Y Cf. A014538, A104338, A153069, A153070, A054543, A118323, A294970/A294971.

%K nonn,cons,easy

%O 0,1

%A _N. J. A. Sloane_

%E More terms from Larry Reeves (larryr(AT)acm.org), May 28 2002

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)