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

%I A053002 #33 Jul 02 2025 16:01:59
%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,
%T A053002 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 A053002 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 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_{t=0..1}(1/sqrt(1-t^4)).
%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="/A053002/b053002.txt">Table of n, a(n) for n = 0..19999</a>
%H A053002 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GausssConstant.html">Gauss's Constant</a>
%H A053002 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A053002 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H A053002 OEIS Wiki, <a href="/wiki/Gauss&#39;s_constant">Gauss's constant</a>
%e A053002 0.83462684167407318628142973...
%t A053002 ContinuedFraction[1/ArithmeticGeometricMean[1, Sqrt[2]] , 100]  (* _Jean-François Alcover_, Apr 18 2011 *)
%o A053002 (PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(1/agm(1, sqrt(2))); for (n=1, 20000, write("b053002.txt", n-1, " ", x[n])); } \\ _Harry J. Smith_, Apr 20 2009
%Y A053002 Cf. A014549 (decimal expansion).
%K A053002 nonn,cofr,nice,easy
%O A053002 0,3
%A A053002 _N. J. A. Sloane_, Feb 21 2000
%E A053002 More terms from _James Sellers_, Feb 22 2000
%E A053002 Offset changed by _Andrew Howroyd_, Aug 03 2024