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 A219011 #22 Jul 20 2025 08:59:30
%S A219011 5,15005,792070839820228500005
%N A219011 Denominators in a product expansion for sqrt(5).
%C A219011 Apart from the initial term same as A145275.
%C A219011 a(3) has 105 digits and a(4) has 523 digits.
%C A219011 The product expansion in question is sqrt(5) = Product_{n >= 0} (1 + 2*A219010(n)/A219011(n)) = (1 + 6/5)*(1 + 246/15005)*(1 + 56287506246/792070839820228500005)*....
%H A219011 Amiram Eldar, <a href="/A219011/b219011.txt">Table of n, a(n) for n = 0..4</a>
%F A219011 a(n) = Fibonacci(5^(n+1))/Fibonacci(5^n).
%F A219011 a(n) = A219010(n)^2 - A219010(n) - 1.
%F A219011 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) = 5.
%F A219011 a(n) = Lucas(4*5^n) - Lucas(2*5^n) + 1. - _Ehren Metcalfe_, Jul 29 2017
%t A219011 a[n_] := LucasL[4*5^n] - LucasL[2*5^n] + 1; Array[a, 3, 0] (* _Amiram Eldar_, Jul 20 2025 *)
%o A219011 (Maxima) A219011(n):=fib(5^(n+1))/fib(5^n)$
%o A219011 makelist(A219011(n),n,0,3);
%Y A219011 Cf. A000032, A000045, A145275, A219010, A219013, A219015.
%K A219011 nonn,easy
%O A219011 0,1
%A A219011 _Peter Bala_, Nov 09 2012