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 A219013 #16 Jul 20 2025 08:59:39
%S A219013 11,523451,39571031999225940638473470251
%N A219013 Denominators in a product expansion for sqrt(3).
%C A219013 The product expansion in question is sqrt(3) = Product_{n >= 0} (1 + 2*A219012(n)/A219013(n)) = (1 + 2*4/11)*(1 + 2*724/523451)*(1 + 2*198924689265124/39571031999225940638473470251)*....
%H A219013 Amiram Eldar, <a href="/A219013/b219013.txt">Table of n, a(n) for n = 0..4</a>
%F A219013 Let alpha = 1/2*(sqrt(2) + sqrt(6)) and put f(n) = 1/sqrt(6)*{alpha^n - (-1/alpha)^n}. Then a(n) = f(5^(n+1))/f(5^n).
%F A219013 a(n) = A219012(n)^2 - A219012(n) - 1.
%F A219013 Recurrence equation: a(n+1) = 5/2*(a(n)^4 - a(n)^2)*sqrt(4*a(n) + 5) + a(n)^5 + 15/2*a(n)^4 - 25/2*a(n)^2 + 5 with initial condition a(0) = 11.
%t A219013 RecurrenceTable[{a[n+1] == 5/2*(a[n]^4 - a[n]^2)*Sqrt[4*a[n] + 5] + a[n]^5 + 15/2*a[n]^4 - 25/2*a[n]^2 + 5, a[0] == 11}, a, {n, 0, 3}] (* _Amiram Eldar_, Jul 20 2025 *)
%Y A219013 Cf. A188887, A219011, A219012, A219015.
%K A219013 nonn,easy
%O A219013 0,1
%A A219013 _Peter Bala_, Nov 09 2012