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 A089603 #24 Feb 16 2025 08:32:51
%S A089603 1,2,8,40,226,1380,8880,59280,406416,2842400,20186752,145119616,
%T A089603 1053575336,7711639760,56834201280,421327859520,3139306406850,
%U A089603 23494847031300,176526280319120,1330929290036560,10065855468854980,76341682531733960,580460500453098080
%N A089603 G.f.: sqrt(1/agm(1, 1-8*x)) = sqrt(o.g.f. for A081085).
%H A089603 Seiichi Manyama, <a href="/A089603/b089603.txt">Table of n, a(n) for n = 0..500</a>
%H A089603 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Arithmetic-GeometricMean.html">Arithmetic-geometric mean</a>
%H A089603 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean">Arithmetic-geometric mean</a>
%F A089603 a(n) ~ 2^(3*n - 1/2) / (n * sqrt(Pi*log(n))) * (1 - (gamma/2 + log(2))/log(n) + (3*gamma^2/8 + 3*log(2)*gamma/2 + 3*log(2)^2/2 - Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 29 2019
%t A089603 CoefficientList[Series[Sqrt[2*EllipticK[1/(1 - 1/(4*x))^2]/(Pi*(1 - 4*x))], {x, 0, 25}], x] (* _Vaclav Kotesovec_, Sep 26 2019 *)
%t A089603 nmax = 25; CoefficientList[Series[Sqrt[Hypergeometric2F1[1/2, 1/2, 1, 16*x*(1 - 4*x)]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 26 2019 *)
%o A089603 (PARI) Vec( 1/agm(1,1-8*x+O(x^66))^(1/2) ) \\ _Joerg Arndt_, Aug 14 2013
%Y A089603 Cf. A081085, A089602.
%K A089603 nonn
%O A089603 0,2
%A A089603 _N. J. A. Sloane_, Dec 31 2003