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A257096 Decimal expansion of I3(u,v) = A248897/AG3(u,v) for u=1, v=2. 3


%S 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,

%T 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,

%U 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

%N Decimal expansion of I3(u,v) = A248897/AG3(u,v) for u=1, v=2.

%C 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".

%C 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)).

%H Stanislav Sykora, <a href="/A257096/b257096.txt">Table of n, a(n) for n = 0..2000</a>

%H J. M. Borwein, P. B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Transactions of the AMS, 323 (1991), 691-701.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Arithmetic-GeometricMean.html">Arithmetic-Geometric Mean</a>, Equations 26-32.

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

%e 0.724235633800971429538923331111501838330976344683295530...

%t 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 *)

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

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

%o while(1, anext=(an+2*bn)/3;

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

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

%o return((2*Pi/(3*sqrt(3)))/an);}

%o a = I3(1,2)

%Y Cf. A248897, A257097.

%K nonn,cons

%O 0,1

%A _Stanislav Sykora_, Apr 16 2015

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Last modified January 16 06:59 EST 2019. Contains 319188 sequences. (Running on oeis4.)