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 A342813 #14 Feb 05 2025 00:15:31 %S A342813 4,3,1,4,0,7,1,2,5,4,6,6,7,7,2,9,5,0,3,3,0,2,2,9,1,9,8,6,4,1,6,3,0,9, %T A342813 3,7,3,0,0,9,2,6,6,3,4,2,2,4,7,6,6,2,7,8,6,3,6,5,4,4,0,3,7,7,7,2,9,8, %U A342813 2,9,0,3,4,1,7,4,0,3,6,3,9,6,1,3,1,3,4 %N A342813 Decimal expansion of the limit of AGM(1, 2, ..., n)/n. %C A342813 The two-parameter arithmetic-geometric mean function AGM is defined by taking the limit of the sequence of iterates of the map (x, y) -> ((x+y)/2, sqrt(x*y)). This can be extended to an arbitrary finite sequence of numbers by defining AGM(x(1), ..., x(n)) = AGM((x(1)+...+x(n))/n, (x(1)*...*x(n))^(1/n)). Different extensions of the definition to more than two parameters are also possible, such as the one used in A332093. %F A342813 Equals Pi/4 * (1/2 + 1/e) / K(((e-2)/(e+2))^2) where K is the complete elliptic integral of the first kind. %e A342813 0.431407125466772950330229198641630937300926634224766278636544... %t A342813 RealDigits[Pi/4 * (1/2 + 1/E) / EllipticK[((E-2)/(E+2))^2], 10, 100][[1]] %t A342813 RealDigits[ArithmeticGeometricMean[(2 + E)/(4 E), 1/Sqrt[2 E]], 10, 100][[1]] (* _Jan Mangaldan_, Dec 07 2021 *) %o A342813 (PARI) Pi/4 * (exp(-1)+.5) / ellK(1-4/(exp(1)+2)) \\ _Charles R Greathouse IV_, Feb 05 2025 %o A342813 (PARI) agm(exp(-1)/2+1/4, exp(-1/2)/sqrt(2)) \\ _Charles R Greathouse IV_, Feb 05 2025 %Y A342813 Cf. A332093. %K A342813 nonn,cons %O A342813 0,1 %A A342813 _Ben Whitmore_, Mar 22 2021