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 A053002 Continued fraction for 1 / M(1,sqrt(2)) (Gauss's constant). 4
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS On May 30, 1799, Gauss discovered that this number is also equal to (2/Pi)*Integral_{t=0..1}(1/sqrt(1-t^4)). 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). REFERENCES J. M. Borwein and P. B. Borwein, Pi and the AGM, page 5. J. R. Goldman, The Queen of Mathematics, 1998, p. 92. LINKS Harry J. Smith, Table of n, a(n) for n = 1..20000 Eric Weisstein's World of Mathematics, Gauss's Constant G. Xiao, Contfrac OEIS Wiki, Gauss's constant EXAMPLE 0.83462684167407318628142973... MATHEMATICA ContinuedFraction[1/ArithmeticGeometricMean[1, Sqrt[2]] , 100]  (* Jean-François Alcover, Apr 18 2011 *) PROG (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 CROSSREFS Cf. A014549. Sequence in context: A156148 A224867 A156824 * A053003 A167202 A204914 Adjacent sequences:  A052999 A053000 A053001 * A053003 A053004 A053005 KEYWORD nonn,cofr,nice,easy AUTHOR N. J. A. Sloane, Feb 21 2000 EXTENSIONS More terms from James A. Sellers, Feb 22 2000 STATUS approved

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Last modified May 30 05:35 EDT 2020. Contains 334712 sequences. (Running on oeis4.)