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 A053004 Decimal expansion of AGM(1,sqrt(2)). 15
 1, 1, 9, 8, 1, 4, 0, 2, 3, 4, 7, 3, 5, 5, 9, 2, 2, 0, 7, 4, 3, 9, 9, 2, 2, 4, 9, 2, 2, 8, 0, 3, 2, 3, 8, 7, 8, 2, 2, 7, 2, 1, 2, 6, 6, 3, 2, 1, 5, 6, 5, 1, 5, 5, 8, 2, 6, 3, 6, 7, 4, 9, 5, 2, 9, 4, 6, 4, 0, 5, 2, 1, 4, 1, 4, 3, 9, 1, 5, 6, 7, 0, 8, 3, 5, 8, 8, 5, 5, 5, 6, 4, 8, 9, 7, 9, 3, 3, 8, 9, 3, 7, 5, 9, 0 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS AGM(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 George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195. 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 = 1..20000 Eric Weisstein's World of Mathematics, Gauss's Constant. FORMULA Equals Pi/(2*A085565). - Nathaniel Johnston, May 26 2011 Equals Integral_{x=0..Pi/2} sqrt(sin(x)) or Integral_{x=0..1} sqrt(x/(1-x^2)). - Jean-François Alcover, Apr 29 2013 [cf. Boros & Moll p. 195] Equals Product_{n>=1} (1+1/A033566(n)) and also 2*AGM(1, i)/(1+i)) where i is the imaginary unit. - Dimitris Valianatos, Oct 03 2016 Conjecturally equals 1/( Sum_{n = -infinity..infinity} exp(-Pi*(n+1/2)^2 ) )^2. Cf. A096427. - Peter Bala, Jun 10 2019 From Amiram Eldar, Aug 26 2020: (Start) Equals 2 * A076390. Equals Integral_{x=0..Pi} sin(x)^2/sqrt(1 + sin(x)^2) dx. (End) Equals sqrt(2/Pi)*Gamma(3/4)^2 = Integral_{x = 0..1} 1/(1 - x^2)^(1/4) dx = Pi/Integral_{x = 0..1} 1/(1 - x^2)^(3/4) dx. - Peter Bala, Jan 05 2022 From Peter Bala, Mar 02 2022: (Start) Equals 2*Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx. Equals 1 - Integral_{x = 0..1} (sqrt(1 - x^4) - 1)/x^2 dx. Equals hypergeom([-1/2, -1/4], [3/4], 1) = 1 + Sum_{n >= 0} 1/(4*n + 3)*Catalan(n)*(1/2^(2*n+1)). Cf. A096427. (End) EXAMPLE 1.19814023473559220743992249228... MATHEMATICA RealDigits[ N[ ArithmeticGeometricMean[1, Sqrt[2]], 105]][[1]] (* Jean-François Alcover, Jan 30 2012 *) RealDigits[N[(1+Sqrt[2])Pi/(4EllipticK[17-12Sqrt[2]]), 105]][[1]] (* Jean-François Alcover, Jun 02 2019 *) PROG (PARI) default(realprecision, 20080); x=agm(1, sqrt(2)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b053004.txt", n, " ", d)) \\ Harry J. Smith, Apr 20 2009 (PARI) 2*real(agm(1, I)/(1+I)) \\ Michel Marcus, Jul 26 2018 (Python) from mpmath import mp, agm, sqrt mp.dps=106 print([int(z) for z in list(str(agm(1, sqrt(2))).replace('.', '')[:-1])]) # Indranil Ghosh, Jul 11 2017 CROSSREFS Cf. A014549, A053002, A053003, A076390, A085565, A096427. Sequence in context: A198920 A354593 A244115 * A019888 A154975 A199472 Adjacent sequences: A053001 A053002 A053003 * A053005 A053006 A053007 KEYWORD nonn,cons,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 November 30 11:09 EST 2022. Contains 358441 sequences. (Running on oeis4.)