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 A014549 Decimal expansion of 1 / M(1,sqrt(2)) (Gauss's constant). 13
 8, 3, 4, 6, 2, 6, 8, 4, 1, 6, 7, 4, 0, 7, 3, 1, 8, 6, 2, 8, 1, 4, 2, 9, 7, 3, 2, 7, 9, 9, 0, 4, 6, 8, 0, 8, 9, 9, 3, 9, 9, 3, 0, 1, 3, 4, 9, 0, 3, 4, 7, 0, 0, 2, 4, 4, 9, 8, 2, 7, 3, 7, 0, 1, 0, 3, 6, 8, 1, 9, 9, 2, 7, 0, 9, 5, 2, 6, 4, 1, 1, 8, 6, 9, 6, 9, 1, 1, 6, 0, 3, 5, 1, 2, 7, 5, 3, 2, 4, 1, 2, 9, 0, 6, 7, 8, 5 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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, Wiley, 1987, page 5. J. R. Goldman, The Queen of Mathematics, 1998, p. 92. LINKS Harry J. Smith, Table of n, a(n) for n = 0..20000 Eric Weisstein's World of Mathematics, Gauss's Constant Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean FORMULA Equals (lim_{k -> infinity} p(k))/(1+i) and (lim_{k -> infinity} q(k))/(1+i), where i is the imaginary unit, p(0) = 1, q(0) = i, p(k+1) = 2*p(k)*q(k)/(p(k)+q(k)) and q(k+1) = sqrt(p(k)*q(k)) for k >= 0. - A.H.M. Smeets, Jul 26 2018 Equals the infinite quotient product (3/4)*(6/5)*(7/8)*(10/9)*(11/12)*(14/13)*(15/16)*... . - James Maclachlan, Jul 28 2019 EXAMPLE 0.8346268416740731862814297327990468... MATHEMATICA RealDigits[Gamma[1/4]^2/(2*Pi^(3/2)*Sqrt), 10, 105][] (* or: *) RealDigits[1/ArithmeticGeometricMean[1, Sqrt], 10, 105][] (* Jean-François Alcover, Dec 13 2011, updated Nov 11 2016, after Eric W. Weisstein *) PROG (PARI) default(realprecision, 20080); x=10*agm(1, sqrt(2))^-1; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b014549.txt", n, " ", d)); \\ Harry J. Smith, Apr 20 2009 (PARI) 1/agm(sqrt(2), 1) \\ Charles R Greathouse IV, Feb 04 2015 (PARI) sqrt(Pi/2)/gamma(3/4)^2 \\ Charles R Greathouse IV, Feb 04 2015 (Python) from mpmath import * mp.dps=105 print map(int, list(str(1/agm(sqrt(2)))[2:-1])) # Indranil Ghosh, Jul 11 2017 (MAGMA) SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(Pi(R)/2)/Gamma(3/4)^2; // G. C. Greubel, Aug 17 2018 CROSSREFS Cf. A053002, A053003, A053004, A244644. Sequence in context: A222232 A091895 A111436 * A021549 A013665 A209059 Adjacent sequences:  A014546 A014547 A014548 * A014550 A014551 A014552 KEYWORD nonn,cons,nice AUTHOR EXTENSIONS Extended to 105 terms by Jean-François Alcover, Dec 13 2011 a(104) corrected by Andrew Howroyd, Feb 23 2018 STATUS approved

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Last modified October 22 12:30 EDT 2019. Contains 328318 sequences. (Running on oeis4.)