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 A053002 Continued fraction for 1 / M(1,sqrt(2)) (Gauss's constant). 4

%I

%S 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,

%T 3,1,2,6,1,5,1,1,2,1,13,2,2,5,1,2,2,1,5,1,3,1,3,1,2,2,2,2,8,3,1,2,2,1,

%U 10,2,2,2,3,3,1,7,1,8,3,1,1,1,1,1,1,1,1,5,2,1,2,17,1,4,31,2,2,5,30,1,8,2

%N Continued fraction for 1 / M(1,sqrt(2)) (Gauss's constant).

%C On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral_{t=0..1}(1/sqrt(1-t^4)).

%C 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 J. M. Borwein and P. B. Borwein, Pi and the AGM, page 5.

%D J. R. Goldman, The Queen of Mathematics, 1998, p. 92.

%H Harry J. Smith, <a href="/A053002/b053002.txt">Table of n, a(n) for n = 1..20000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GausssConstant.html">Gauss's Constant</a>

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%H OEIS Wiki, <a href="/wiki/Gauss&#39;s_constant">Gauss's constant</a>

%e 0.83462684167407318628142973...

%t ContinuedFraction[1/ArithmeticGeometricMean[1, Sqrt[2]] , 100] (* _Jean-François Alcover_, Apr 18 2011 *)

%o (PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(1/agm(1, sqrt(2))); for (n=1, 20000, write("b053002.txt", n, " ", x[n])); } \\ _Harry J. Smith_, Apr 20 2009

%Y Cf. A014549.

%K nonn,cofr,nice,easy

%O 1,3

%A _N. J. A. Sloane_, Feb 21 2000

%E More terms from _James A. Sellers_, Feb 22 2000

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Last modified July 18 19:00 EDT 2019. Contains 325144 sequences. (Running on oeis4.)