This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A053004 #113 Jul 02 2025 16:01:59
%S A053004 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,
%T A053004 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,
%U A053004 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
%N A053004 Decimal expansion of AGM(1,sqrt(2)).
%C A053004 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).
%D A053004 George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 195.
%D A053004 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, page 5.
%D A053004 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 420.
%D A053004 J. R. Goldman, The Queen of Mathematics, 1998, p. 92.
%H A053004 Harry J. Smith, <a href="/A053004/b053004.txt">Table of n, a(n) for n = 1..20000</a>
%H A053004 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GausssConstant.html">Gauss's Constant</a>.
%H A053004 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A053004 Equals Pi/(2*A085565). - _Nathaniel Johnston_, May 26 2011
%F A053004 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]
%F A053004 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
%F A053004 Conjecturally equals 1/( Sum_{n = -infinity..infinity} exp(-Pi*(n+1/2)^2 ) )^2. Cf. A096427. - _Peter Bala_, Jun 10 2019
%F A053004 From _Amiram Eldar_, Aug 26 2020: (Start)
%F A053004 Equals 2 * A076390.
%F A053004 Equals Integral_{x=0..Pi} sin(x)^2/sqrt(1 + sin(x)^2) dx. (End)
%F A053004 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
%F A053004 From _Peter Bala_, Mar 02 2022: (Start)
%F A053004 Equals 2*Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx.
%F A053004 Equals 1 - Integral_{x = 0..1} (sqrt(1 - x^4) - 1)/x^2 dx.
%F A053004 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)
%e A053004 1.19814023473559220743992249228...
%p A053004 evalf(GaussAGM(1, sqrt(2)), 144); # _Alois P. Heinz_, Jul 05 2023
%t A053004 RealDigits[ N[ ArithmeticGeometricMean[1, Sqrt[2]], 105]][[1]] (* _Jean-François Alcover_, Jan 30 2012 *)
%t A053004 RealDigits[N[(1+Sqrt[2])Pi/(4EllipticK[17-12Sqrt[2]]), 105]][[1]] (* _Jean-François Alcover_, Jun 02 2019 *)
%o A053004 (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
%o A053004 (PARI) 2*real(agm(1, I)/(1+I)) \\ _Michel Marcus_, Jul 26 2018
%o A053004 (Python)
%o A053004 from mpmath import mp, agm, sqrt
%o A053004 mp.dps=106
%o A053004 print([int(z) for z in list(str(agm(1, sqrt(2))).replace('.', '')[:-1])]) # _Indranil Ghosh_, Jul 11 2017
%Y A053004 Cf. A014549, A053002, A053003, A076390, A085565, A096427.
%K A053004 nonn,cons,nice,easy
%O A053004 1,3
%A A053004 _N. J. A. Sloane_, Feb 21 2000
%E A053004 More terms from _James Sellers_, Feb 22 2000