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%I #112 Sep 03 2023 10:20:52
%S 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,
%T 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,
%U 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
%N Decimal expansion of 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, Wiley, 1987, page 5.
%D J. R. Goldman, The Queen of Mathematics, 1998, p. 92.
%H Harry J. Smith, <a href="/A014549/b014549.txt">Table of n, a(n) for n = 0..20000</a>
%H Markus Faulhuber, <a href="https://doi.org/10.1007/s11139-019-00224-2">An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures</a>, The Ramanujan Journal volume 54, pages 1-27 (2021). See pp. 4 and 24.
%H Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, <a href="https://arxiv.org/abs/2209.04202">The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators</a>, arXiv:2209.04202 [math.CA], 2022, p. 2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GausssConstant.html">Gauss's Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Arithmetic-GeometricMean.html">Arithmetic-Geometric Mean</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F Equals (lim_{k->oo} p(k))/(1+i) and (lim_{k->oo} 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
%F Equals the infinite quotient product (3/4)*(6/5)*(7/8)*(10/9)*(11/12)*(14/13)*(15/16)*... . - _James Maclachlan_, Jul 28 2019
%F Equals (9/15)*hypergeom([1/2, 3/4], [9/4], 1). - _Peter Bala_, Mar 03 2022
%F Equals A062539 / Pi. - _Amiram Eldar_, May 04 2022
%F From _Stefano Spezia_, Sep 29 2022: (Start)
%F Equals theta4(exp(-Pi))^2.
%F Equals sqrt(2)*A093341/Pi. (End)
%F Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)^2/16^k. - _Amiram Eldar_, Jul 04 2023
%F From _Gerry Martens_, Jul 31 2023: (Start)
%F Equals 2*Gamma(5/4)/(sqrt(Pi)*Gamma(3/4)).
%F Equals hypergeom([1/4, -2/4], [1], 1). (End)
%e 0.8346268416740731862814297327990468...
%p evalf(1/GaussAGM(1, sqrt(2)), 144); # _Alois P. Heinz_, Jul 05 2023
%t RealDigits[Gamma[1/4]^2/(2*Pi^(3/2)*Sqrt[2]), 10, 105][[1]] (* or: *)
%t RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2]], 10, 105][[1]] (* _Jean-François Alcover_, Dec 13 2011, updated Nov 11 2016, after _Eric W. Weisstein_ *)
%t First[RealDigits[N[EllipticTheta[4, Exp[-Pi]]^2, 90]]] (* _Stefano Spezia_, Sep 29 2022 *)
%o (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
%o (PARI) 1/agm(sqrt(2),1) \\ _Charles R Greathouse IV_, Feb 04 2015
%o (PARI) sqrt(Pi/2)/gamma(3/4)^2 \\ _Charles R Greathouse IV_, Feb 04 2015
%o (Python)
%o from mpmath import mp, agm, sqrt
%o mp.dps=105
%o print([int(z) for z in list(str(1/agm(sqrt(2)))[2:-1])]) # _Indranil Ghosh_, Jul 11 2017
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(Pi(R)/2)/Gamma(3/4)^2; // _G. C. Greubel_, Aug 17 2018
%Y Cf. A053002, A053003, A053004, A062539, A093341, A244644.
%K nonn,cons,nice
%O 0,1
%A _Eric W. Weisstein_, _N. J. A. Sloane_
%E Extended to 105 terms by _Jean-François Alcover_, Dec 13 2011
%E a(104) corrected by _Andrew Howroyd_, Feb 23 2018
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