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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)

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Last modified May 26 03:22 EDT 2017. Contains 287073 sequences.