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A006752 Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...
(Formerly M4593)
231
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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
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, equation 3:3:7.
LINKS
Milton Abramowitz and Irene A. Stegun, editors, Catalan's constant, Handbook of Mathematical Functions, December 1972, p. 807, 23.2.21 for n=2.
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.
David M. Bradley, Representations of Catalan's constant, 2001.
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.
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
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, 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)
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)
EXAMPLE
0.91596559417721901505460351493238411077414937428167213426649811962176301977...
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 *)
PROG
(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
CROSSREFS
Sequence in context: A205326 A021526 A019791 * A271856 A164802 A201888
KEYWORD
nonn,cons,easy
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 28 2002
STATUS
approved

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Last modified May 29 08:39 EDT 2024. Contains 372926 sequences. (Running on oeis4.)