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A006752 Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...

%I A006752 M4593 #259 Jun 04 2025 00:28:47
%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,
%T A006752 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 A006752 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 A006752 Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...
%C A006752 Usually denoted by G.
%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. See equation 3:3:7 in Spanier and Oldham. - _Harry J. Smith_, May 07 2009
%D A006752 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 57, 554.
%D A006752 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 53-59.
%D A006752 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006752 Jerome Spanier and Keith B. Oldham, An Atlas of Functions, 1987, equations 1:7:3, 3:3:7.
%H A006752 Harry J. Smith, <a href="/A006752/b006752.txt">Table of n, a(n) for n = 0..20000</a>
%H A006752 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 A006752 Victor Adamchik, <a href="https://library.wolfram.com/infocenter/Demos/109/">33 representations for Catalan's constant</a>.
%H A006752 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 A006752 Peter Bala, <a href="/A002117/a002117.pdf">New series for old functions</a>.
%H A006752 David M. Bradley, <a href="https://www.researchgate.net/publication/2325473_Representations_of_Catalan%27s_Constant">Representations of Catalan's constant</a>, 2001.
%H A006752 Eugène-Charles Catalan, <a href="https://www.persee.fr/doc/marb_0776-3123_1867_num_33_1_1705">Mémoire sur la transformation des séries et sur quelques intégrales définies</a>, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
%H A006752 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 A006752 Greg Fee, <a href="https://www.gutenberg.org/ebooks/682">Catalan's Constant to 300000 digits</a>, Project Gutenberg, 1996.
%H A006752 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 A006752 Philippe Flajolet and Ilan Vardi, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">Zeta function expansions of some classical constants</a>.
%H A006752 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 A006752 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 A006752 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 A006752 A. Lupas, <a href="/A006752/a006752.pdf">Formulae for some classical constants</a>, in Proceedings of ROGER-2000, 2000. [Local copy]
%H A006752 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 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>, Project Gutenberg, 1997.
%H A006752 Kh. Hessami Pilehrood and Tatiana Hessami Pilehrood, <a href="https://doi.org/10.46298/dmtcs.504">Series acceleration formulas for beta values</a>, Discr. Math. Theor. Comp. Sci. 12 (2) (2010) 223-236.
%H A006752 Simon Plouffe, <a href="http://www.plouffe.fr/simon/gendev/915965.html">Generalized expansions of real numbers</a>, 2006.
%H A006752 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 A006752 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalansConstant.html">Catalan's Constant</a>.
%H A006752 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalansConstantDigits.html">Catalan's Constant Digits</a>.
%H A006752 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.
%H A006752 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrigammaFunction.html">Trigamma Function</a>
%H A006752 Wikipedia, <a href="http://en.wikipedia.org/wiki/Catalan%27s_constant">Catalan's constant</a>.
%H A006752 Sheldon Yang, <a href="https://doi.org/10.1080/0020739X.1992.10715688">Some properties of Catalan's constant G</a>, Int. J. Math. Educ. Sci. Technol 23 (4) (1992) 549-556.
%H A006752 David G. Zeitoun and Thierry Dana-Picard, <a href="https://arxiv.org/abs/2412.11726">A half-automated study of a 2-parameter family of integrals</a>, arXiv:2412.11726 [math.CO], 2024. See p. 8.
%F A006752 G = Integral_{x=0..1} arctan(x)/x dx.
%F A006752 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 A006752 G = (zeta(2,1/4)- zeta(2,3/4))/16. - _Gerry Martens_, May 27 2011 [With the Hurwitz zeta function zeta.]
%F A006752 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 A006752 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 A006752 G = phi(-1, 2, 1/2)/4 = A247685/4, where phi is Lerch transcendent. - _Jean-François Alcover_, Mar 28 2013
%F A006752 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 A006752 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 A006752 G = (Zeta(2, 1/4) - Pi^2)/8 = (Psi(1, 1/4) - Pi^2)/8 = (A282823-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 A006752 Equals Im(Li_{2}(i)). - _Peter Luschny_, Oct 04 2019
%F A006752 Equals -Integral_{x=0..Pi/4} log(tan(x)) dx. - _Amiram Eldar_, Jun 29 2020
%F A006752 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 A006752 From _Peter Bala_, Dec 08 2021: (Start)
%F A006752 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 A006752 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 A006752 From _Amiram Eldar_, Jan 07 2024: (Start)
%F A006752 Equals beta(2), where beta is the Dirichlet beta function.
%F A006752 Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^2)^(-1). (End)
%F A006752 Equals 2*Integral_{x=0..Pi/4} log(2*cos(x)) dx = -2*Integral_{x=0..Pi/4} log(2*sin(x)) dx (see Finch). - _Stefano Spezia_, Nov 14 2024
%F A006752 Equals Integral_{x=0..Pi/4} log((1 + tan(x))/(1 - tan(x))) dx. - _Kritsada Moomuang_, Jun 03 2025
%e A006752 0.91596559417721901505460351493238411077414937428167213426649811962176301977...
%p A006752 evalf(Catalan) ; # _R. J. Mathar_, Apr 09 2013
%t A006752 nmax = 1000; First[RealDigits[Catalan, 10, nmax]] (* _Stuart Clary_, Dec 17 2008 *)
%t A006752 Integrate[ArcTan[x]/x, {x, 0, 1}] (* _N. J. A. Sloane_, May 03 2013 *)
%t A006752 N[Im[PolyLog[2, I]], 100] (* _Peter Luschny_, Oct 04 2019 *)
%o A006752 (PARI) { mydigits=20000; default(realprecision, mydigits+80); s=1.0; n=5*mydigits; 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, mydigits, 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 A006752 (PARI) default(realprecision,1000+2); /* 1000 terms */
%o A006752 s=sumalt(n=0,(-1)^n/(2*n+1)^2);
%o A006752 v=Vec(Str(s)); /* == ["0", ".", "9", "1", "5", "9", "6", ...*/
%o A006752 vector(#v-2,n,eval(v[n+2]))
%o A006752 /* _Joerg Arndt_, Aug 25 2011 */
%o A006752 (PARI) Catalan \\ _Charles R Greathouse IV_, Nov 20 2011
%o A006752 (PARI) (zetahurwitz(2,1/4)-Pi^2)/8 \\ _Charles R Greathouse IV_, Jan 30 2018
%o A006752 (PARI) lerchphi(-1, 2, 1/2)/4 \\ _Charles R Greathouse IV_, Jan 30 2025
%o A006752 (Magma) R:= RealField(100); Catalan(R); // _G. C. Greubel_, Aug 21 2018
%Y A006752 Cf. A014538, A104338, A153069, A153070, A054543, A118323, A294970/A294971.
%K A006752 nonn,cons,easy
%O A006752 0,1
%A A006752 _N. J. A. Sloane_
%E A006752 More terms from Larry Reeves (larryr(AT)acm.org), May 28 2002