A247685 -id:A247685 - cp's OEIS frontend

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

N. J. A. Sloane

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

			0.91596559417721901505460351493238411077414937428167213426649811962176301977...

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.

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.

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

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

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

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 *)

{ 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 + ...  */

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 */

Catalan \\ Charles R Greathouse IV, Nov 20 2011

(zetahurwitz(2,1/4)-Pi^2)/8 \\ Charles R Greathouse IV, Jan 30 2018

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

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

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

A218387 Decimal expansion of the spanning tree constant of the square lattice.

Original entry on oeis.org

1, 1, 6, 6, 2, 4, 3, 6, 1, 6, 1, 2, 3, 2, 7, 5, 1, 2, 0, 5, 5, 3, 5, 3, 7, 8, 2, 5, 8, 7, 3, 5, 7, 9, 6, 7, 5, 4, 5, 6, 2, 6, 4, 6, 1, 5, 9, 4, 3, 3, 4, 9, 0, 8, 1, 0, 4, 4, 0, 0, 6, 2, 7, 6, 4, 4, 6, 9, 9, 0, 5, 4, 7, 5, 2, 1, 7, 5, 5, 4, 4, 6, 9, 0, 6, 5, 0, 7, 2, 9, 7, 2, 1, 2, 5, 3, 6, 2, 3, 5, 6, 3, 5, 8, 9, 1, 2, 1, 1, 1, 1, 5, 1

Offset: 1

R. J. Mathar, Oct 27 2012

			1.16624361612327512055353782587357967545626461594...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.7 and 5.22.6, pp. 54, 399.
Asmus L. Schmidt, Ergodic theory of complex continued fractions, Number Theory with an Emphasis on the Markoff Spectrum, in: A. D. Pollington and W. Moran (eds.), Number Theory with an Emphasis on the Markoff Spectrum, Dekker, 1993, pp. 215-226.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Anthony J. Guttmann, Spanning tree generating functions and Mahler measure, arXiv:1207.2815 [math-ph], 2012.
Sheldon Yang, Some properties of Catalan's constant G, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556, L*(1).

Cf. A006752 (Catalan), A088538 (4/Pi), A229728, A247685.

R:= RealField(100); 4*Catalan(R)/Pi(R); // G. C. Greubel, Aug 23 2018

Maple
```
evalf(Catalan*4/Pi) ;
```

RealDigits[4*Catalan/Pi, 10, 100][[1]] (* G. C. Greubel, Aug 23 2018 *)

default(realprecision, 100); 4*Catalan/Pi \\ G. C. Greubel, Aug 23 2018

Equals the product of A006752 by A088538.
From Amiram Eldar, Jul 22 2020: (Start)
Equals 1 + Sum_{k>=1} (2*k-1)!!^2/((2*k)!!^2 * (2*k + 1)).
Equals Sum_{k>=0} binomial(2*k,k)^2/(16^k * (2*k + 1)). (End)
Equals (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)^2) / (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)) [Schmidt] (see Finch). - Stefano Spezia, Nov 07 2024
Equals log(A229728) = A247685/Pi. - Hugo Pfoertner, Nov 07 2024
Equals Integral_{x=0..1} EllipticK(x)/(Pi*sqrt(x)) dx. - Kritsada Moomuang, Jun 21 2025

A221209 Decimal expansion of two times the Catalan constant.

Original entry on oeis.org

1, 8, 3, 1, 9, 3, 1, 1, 8, 8, 3, 5, 4, 4, 3, 8, 0, 3, 0, 1, 0, 9, 2, 0, 7, 0, 2, 9, 8, 6, 4, 7, 6, 8, 2, 2, 1, 5, 4, 8, 2, 9, 8, 7, 4, 8, 5, 6, 3, 3, 4, 4, 2, 6, 8, 5, 3, 2, 9, 9, 6, 2, 3, 9, 2, 4, 3, 5, 2, 6, 0, 3, 9, 5, 5, 2, 5, 0, 9, 5, 3, 8, 9, 5, 8, 7, 1, 3, 0, 2, 5, 8, 5, 2, 2, 3, 0, 2, 1, 2

Offset: 1

R. J. Mathar, Feb 21 2013

			1.83193118835443803010920702986476822154...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.
I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, 5th edition, Academic Press, 1994, eq. (3.521.2).

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. D. Krupnikov and K. S. Kölbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 594.

Cf. A006752, A247685.

SetDefaultRealField(RealField(100)); R:=RealField(); 2*Catalan(R); // G. C. Greubel, Aug 25 2018

Maple
```
evalf(2*Catalan) ;
```

RealDigits[2 Catalan, 10, 100][[1]] (* Bruno Berselli, Feb 21 2013 *)

default(realprecision, 100); 2*Catalan \\ G. C. Greubel, Aug 25 2018

Equals Integral_{x=0..oo} x/cosh(x) dx.
Equals 2*A006752.
From Amiram Eldar, Aug 20 2020: (Start)
Equals Integral_{x=0..Pi/2} x/sin(x) dx.
Equals 1 + Integral_{x=0..oo} x * exp(-x) * tanh(x) dx. (End)
Equals 3F2(1/2,1,1;3/2,3/2;1) [Krupnikov]. - R. J. Mathar, May 13 2024
From Stefano Spezia, Nov 12 2024: (Start)
Equals Integral_{x=0..oo} arctan(x)/(x*sqrt(x^2 + 1)) dx = Integral_{x=0..1} K(x^2) dx, where K(x) is the complete elliptic integral of the first kind (see Shamos).
Equals Sum_{k>=0} 2^(2*k)/((2*k + 1)^2*binomial(2*k,k)) (see Finch). (End)
Equals A247685/2. - Hugo Pfoertner, Nov 12 2024
Equals Sum_{n>=1} H(2*n) * binomial(2*n, n) / (4^n * (2*n + 1)), where H(n) is the n-th harmonic number. - Antonio Graciá Llorente, Apr 04 2025
Equals Integral_{x=-1..1} -log(abs(x))/(1 + x^2) dx. - Kritsada Moomuang, May 28 2025

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

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A218387 Decimal expansion of the spanning tree constant of the square lattice.

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A221209 Decimal expansion of two times the Catalan constant.

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A263354 Decimal expansion of the generalized hypergeometric function 3F2(1/2,3/2,3/2; 5/2,5/2;x) at x=1/2.

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A375392 Decimal expansion of the hyperbolic volume of the link complement of the Borromean rings.

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