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 A083696 #20 Jan 15 2022 09:47:17
%S A083696 1,6,56,6016,72318976,10460064284409856,
%T A083696 218825889667954898996994670329856,
%U A083696 95769539977943941232017762100658986141884645207653888255921750016
%N A083696 a(n) = Sum_{r=0..2^(n-1)} (5^r/(2r)!)*Product_{k=0..2r-1} (2^n - k).
%C A083696 Similar to A081459: a(n) is the numerator of the same mapping f(r) = (1/2)*(r + 5/r) but with initial value r=1.
%H A083696 G. C. Greubel, <a href="/A083696/b083696.txt">Table of n, a(n) for n = 0..10</a>
%F A083696 a(n)/A083697(n) converges to sqrt(5).
%F A083696 a(n) = a(n-1)^2 + 5*A083697(n-1)^2.
%F A083696 a(n) = 2^(2^n - 1) * Lucas(2^n). - _Vaclav Kotesovec_, Jan 08 2021
%t A083696 Table[Sum[Product[2^n - k, {k, 0, 2*r - 1}]5^r/(2*r)!, {r, 0, 2^(n - 1)}], {n, 0, 8}]
%t A083696 Table[2^(2^n - 1)*LucasL[2^n], {n, 0, 8}] (* _Vaclav Kotesovec_, Jan 08 2021 *)
%o A083696 (Sage) [2^(2^n -1)*lucas_number2(2^n, 1, -1) for n in (0..8)] # _G. C. Greubel_, Jan 14 2022
%o A083696 (Magma) [2^(2^n -1)*Lucas(2^n): n in [0..8]]; // _G. C. Greubel_, Jan 14 2022
%Y A083696 Cf. A000032, A083697, A081459.
%K A083696 easy,nonn
%O A083696 0,2
%A A083696 Mario Catalani (mario.catalani(AT)unito.it), May 22 2003