%I A053002
%S A053002 0,1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,8,36,1,2,5,2,1,1,2,2,6,9,1,1,1,3,
%T A053002 1,2,6,1,5,1,1,2,1,13,2,2,5,1,2,2,1,5,1,3,1,3,1,2,2,2,2,8,3,1,2,2,1,10,
%U A053002 2,2,2,3,3,1,7,1,8,3,1,1,1,1,1,1,1,1,5,2,1,2,17,1,4,31,2,2,5,30,1,8,2
%N A053002 Continued fraction for 1 / M(1,sqrt(2)) (Gauss's constant).
%C A053002 On May 30, 1799, Gauss discovered that this number is also equal to (2/
               Pi)*Integral(1/sqrt(1-t^4),t=0..1).
%C A053002 M(a,b) is the limit of the arithmetic-geometric mean iteration applied 
               repeatedly starting with a and b: a_0=a, b_0=b, a_{n+1}=(a_n+b_n)/
               2, b_{n+1}=sqrt(a_n*b_n).
%D A053002 J. M. Borwein and P. B. Borwein, Pi and the AGM, page 5.
%D A053002 J. R. Goldman, The Queen of Mathematics, 1998, p. 92.
%H A053002 Harry J. Smith, <a href="b053002.txt">Table of n, a(n) for n=1,...,20000</
               a>
%H A053002 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GaussConstant.html">Link to a section of The World of Mathematics.</
               a>
%H A053002 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">
               Contfrac</a>
%H A053002 <a href="Sindx_Con.html#confC">Index entries for continued fractions 
               for constants</a>
%e A053002 0.83462684167407318628142973...
%o A053002 (PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(1/
               agm(1, sqrt(2))); for (n=1, 20000, write("b053002.txt", n, " ", x[n])); 
               } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 20 2009]
%Y A053002 Cf. A014549.
%Y A053002 Sequence in context: A002030 A156148 A156824 this_sequence A053003 A167202 
               A165204
%Y A053002 Adjacent sequences: A052999 A053000 A053001 this_sequence A053003 A053004 
               A053005
%K A053002 nonn,cofr,nice,easy
%O A053002 1,3
%A A053002 N. J. A. Sloane (njas(AT)research.att.com), Feb 21 2000
%E A053002 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 22 2000