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

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

Offset: 0

Author

Stanislav Sykora, Apr 16 2015

Keywords

• nonn
• cons

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)).

Examples

			0.724235633800971429538923331111501838330976344683295530...
		

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.

Crossrefs

Cf. A248897, A257097.

Programs

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 *)

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)
				

Formula

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