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 A378683 #26 Dec 29 2024 14:08:33 %S A378683 1,3,153,21489003,59538796254981950751153, %T A378683 1266343134315970349117919634635229303292221774134557782012151266098003 %N A378683 a(0) = 1, a(n+1) = 6*a(n)^3 - 3*a(n). %C A378683 If we define u(0) = 1 , u(n+1) = (u(n)/3)*(u(n)^2+9) / (u(n)^2 + 1), then u(n) = A238799(n) / a(n) ; this is Halley's method to calculate sqrt(3). %H A378683 Wikipedia, <a href="https://en.wikipedia.org/wiki/Halley's_method">Halley's method</a>. %F A378683 a(n) = ((1 + sqrt(3))^(3^n) - (1 - sqrt(3))^(3^n))/2^((3^n+1)/2) / sqrt(3). %F A378683 a(n) = A002530(3^n). %F A378683 a(n) = A002605(3^n)/2^((3^n+1)/2). %p A378683 a:=1 : A:=NULL : for k to 5 do a:=6*a^3-3*a : A:=A,a od : A; %t A378683 NestList[6#^3-3# &, 1, 5] (* _Stefano Spezia_, Dec 06 2024 *) %Y A378683 Cf. A002530, A002605, A238799. %K A378683 nonn %O A378683 0,2 %A A378683 _Robert FERREOL_, Dec 03 2024