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A006752 Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...
(Formerly M4593)
76
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

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. This is from "An Atlas Of Functions" by Spanier, J. and Oldham, K. B. 1987, equation 3:3:7. */ [Harry J. Smith, May 07 2009]

REFERENCES

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.

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

Yasuyuki Kachi and Pavlos Tzermias, Infinite products involving zeta(3) and Catalan's constant, Journal of Integer Sequences, Vol. 15 (2012), #12.9.4.

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

LINKS

Harry J. Smith, Table of n, a(n) for n=0..20000

Victor Adamchik, 33 representations for Catalan's constant

Greg Fee, Project Gutenberg, Catalan's Constant to 300000 digits

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants

F. M. S. Lima, A rapidly converging Ramanujan-type series for Catalan's constant, arXiv:1207.3139v1 [math.NT], July 13, 2012.

A. Lupas, Formulae for some classical constants, in Proceedings of ROGER-2000, 2000.

T. Papanikolaou and G. Fee, Catalan's Constant [Ramanujan's Formula] to 1,500,000 places [Gutenberg Project Etext]

Simon Plouffe, Generalized expansions of real numbers, 2006.

Eric Weisstein's World of Mathematics, Catalan's Constant

Eric Weisstein's World of Mathematics, Catalan's Constant Digits

Wikipedia, Catalan's constant

FORMULA

c = integrate(x=0..1, arctan(x)/x ).

c = integrate(x=0..1, 3*arctan(x*(1-x)/(2-x))/x ). - Posting to Number Theory List by James McLaughlin, Sep 27 2007

c = (zeta(2,1/4)- zeta(2,3/4))/16 - [Gerry Martens, May 27 2011]

c = 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).

c = -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).

c = phi(-1, 2, 1/2)/4, where phi is Lerch transcendent. [Jean-François Alcover, Mar 28 2013]

c = 1/2*integrate(x=0..Pi/2, log(cot(x)+csc(x)) ). [Jean-François Alcover, Apr 11 2013]

EXAMPLE

0.9159655941772190150546035149323841107741493742816721342664981196217630197762\

547694793565129261151...

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*)

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

CROSSREFS

Cf. A014538, A104338, A014538, A153069, A153070, A054543, A118323.

Sequence in context: A205326 A021526 A019791 * A164802 A201888 A185825

Adjacent sequences:  A006749 A006750 A006751 * A006753 A006754 A006755

KEYWORD

nonn,cons,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

STATUS

approved

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Last modified October 25 13:53 EDT 2014. Contains 248541 sequences.