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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

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

Examples

			0.91596559417721901505460351493238411077414937428167213426649811962176301977...
		

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 57, 554.
• Steven 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).
• Jerome Spanier and Keith B. Oldham, An Atlas of Functions, 1987, equations 1:7:3, 3:3:7.

Links

• 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, 2001.
• Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
• 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 Gutenberg, 1996.
• 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.
• Philippe Flajolet and Ilan 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]
• David Naccache and Ofer Yifrach-Stav, On Catalan Constant Continued Fractions, arXiv:2210.15669 [cs.SC], 2022.
• T. Papanikolaou and G. Fee, Catalan's Constant [Ramanujan's Formula] to 1,500,000 places, Project Gutenberg, 1997.
• Kh. Hessami Pilehrood and Tatiana Hessami Pilehrood, Series acceleration formulas for beta values, Discr. Math. Theor. Comp. Sci. 12 (2) (2010) 223-236.
• 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, pp. 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.
• Sheldon Yang, Some properties of Catalan's constant G, Int. J. Math. Educ. Sci. Technol 23 (4) (1992) 549-556.
• David G. Zeitoun and Thierry Dana-Picard, A half-automated study of a 2-parameter family of integrals, arXiv:2412.11726 [math.CO], 2024. See p. 8.

Crossrefs

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

Programs

Magma

R:= RealField(100); Catalan(R); // G. C. Greubel, Aug 21 2018

Maple

evalf(Catalan) ; # R. J. Mathar, Apr 09 2013

Mathematica

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 *)

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 + ...  */

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

PARI

lerchphi(-1, 2, 1/2)/4 \\ Charles R Greathouse IV, Jan 30 2025

Formula

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 Mc Laughlin, 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 = A247685/4, 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 = (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

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)

From Amiram Eldar, Jan 07 2024: (Start)

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

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

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

Equals Integral_{x=0..Pi/4} log((1 + tan(x))/(1 - tan(x))) dx. - Kritsada Moomuang, Jun 03 2025

Extensions

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