login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A257096 Decimal expansion of I3(u,v) = A248897/AG3(u,v) for u=1, v=2. 3
7, 2, 4, 2, 3, 5, 6, 3, 3, 8, 0, 0, 9, 7, 1, 4, 2, 9, 5, 3, 8, 9, 2, 3, 3, 3, 1, 1, 1, 1, 5, 0, 1, 8, 3, 8, 3, 3, 0, 9, 7, 6, 3, 4, 4, 6, 8, 3, 2, 9, 5, 5, 3, 0, 4, 9, 8, 9, 2, 4, 7, 6, 0, 7, 2, 5, 1, 1, 4, 3, 5, 6, 4, 7, 3, 6, 3, 5, 5, 8, 5, 5, 2, 3, 5, 8, 4, 6, 2, 2, 3, 9, 6, 1, 3, 9, 4, 0, 3, 8, 9, 3, 8, 5, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

For positive u and v, AG3(u,v) is defined as the common limit of u_k, v_k such that u_0=u, v_0=v, u_(k+1)=(u_k+2*v_k)/3, v_(k+1)=(v_k*(u_k*u_k+u_k*v_k+v_k*v_k)/3)^(1/3). Since the iterative algorithm is similar to that for AGM, AG3 is sometimes referred to as "cubic AGM".

An alternative definition of I3(u,v) is by means of the definite integral I3(u,v) = Integral[x=0,inf](x/((u^3+x^3)*(v^3+x^3)^2)^(1/3)).

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..2000

J. M. Borwein, P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Transactions of the AMS, 323 (1991), 691-701.

Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean, Equations 26-32.

FORMULA

Equals Integral[x=0,inf](x/((1+x^3)*(8+x^3)^2)^(1/3)).

EXAMPLE

0.724235633800971429538923331111501838330976344683295530...

MATHEMATICA

RealDigits[ NIntegrate[(x/((1 + x^3) (8 + x^3)^2)^(1/3)), {x, 0, Infinity}, AccuracyGoal -> 111, WorkingPrecision -> 111]][[1]] (* Robert G. Wilson v, Apr 16 2015 *)

PROG

(PARI) I3(u, v)={my(an=u+0.0, bn=v+0.0, anext=0.0, ncyc=0,

  eps=2*10^(-default(realprecision)));

  while(1, anext=(an+2*bn)/3;

    bn=(bn*(an*an+an*bn+bn*bn)/3)^(1/3); an=anext;

    ncyc++; if((ncyc>3)&&(abs(an-bn)<eps), break));

  return((2*Pi/(3*sqrt(3)))/an); }

a = I3(1, 2)

CROSSREFS

Cf. A248897, A257097.

Sequence in context: A021584 A021062 A176436 * A121562 A167902 A154174

Adjacent sequences:  A257093 A257094 A257095 * A257097 A257098 A257099

KEYWORD

nonn,cons

AUTHOR

Stanislav Sykora, Apr 16 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 26 02:56 EDT 2017. Contains 287073 sequences.