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A257097 Decimal expansion of I3(u,v) = A248897/AG3(u,v) for u=2, v=1. 3
9, 1, 0, 0, 7, 6, 2, 7, 2, 8, 9, 6, 6, 4, 4, 9, 9, 4, 5, 9, 3, 5, 6, 4, 3, 3, 4, 7, 1, 4, 6, 3, 0, 2, 0, 7, 5, 4, 2, 2, 9, 2, 7, 9, 7, 5, 4, 1, 4, 8, 8, 0, 8, 1, 3, 6, 5, 2, 5, 9, 0, 4, 5, 9, 6, 5, 8, 1, 4, 1, 1, 1, 3, 2, 3, 7, 4, 6, 6, 2, 8, 2, 4, 3, 5, 9, 8, 0, 0, 8, 5, 1, 7, 9, 5, 2, 2, 1, 2, 8, 1, 6, 3, 7, 1 (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/((8+x^3)*(1+x^3)^2)^(1/3)).

EXAMPLE

0.9100762728966449945935643347146302075422927975414880813652590...

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(2, 1)

CROSSREFS

Cf. A248897, A257096.

Sequence in context: A249489 A166734 A155783 * A256667 A175764 A269948

Adjacent sequences:  A257094 A257095 A257096 * A257098 A257099 A257100

KEYWORD

nonn,cons

AUTHOR

Stanislav Sykora, Apr 16 2015

STATUS

approved

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Last modified August 19 19:25 EDT 2017. Contains 290821 sequences.