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 A166516 #23 Aug 03 2024 01:50:55
%S A166516 1,1,2,4,10,25,65,169,442,1156,3026,7921,20737,54289,142130,372100,
%T A166516 974170,2550409,6677057,17480761,45765226,119814916,313679522,
%U A166516 821223649,2149991425,5628750625,14736260450,38580030724,101003831722
%N A166516 A product of consecutive doubled Fibonacci numbers.
%H A166516 G. C. Greubel, <a href="/A166516/b166516.txt">Table of n, a(n) for n = 0..500</a>
%H A166516 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).
%F A166516 G.f.: (1-2*x-x^2+x^3) / ( (1-x)*(1+x)*(1-3*x+x^2) ).
%F A166516 a(n) = Fibonacci(2*floor(n/2) + 1)*Fibonacci(2*floor((n-1)/2) + 1).
%F A166516 a(n) = Fibonacci(A166514(2*n))^2 + Fibonacci(A166514(2*n+1))^2.
%F A166516 a(n) = Fibonacci(n)^2 * (1-(-1)^n)/2 + Fibonacci(n-1)*Fibonacci(n+1) * (1+(-1)^n)/2.
%F A166516 a(n+1)*a(n+3) - a(n+2)^2 = Fibonacci(n+2)^2 * (1-(-1)^n)/2.
%F A166516 a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - _G. C. Greubel_, May 15 2016
%t A166516 CoefficientList[Series[(1-2x-x^2+x^3)/((1-x)(1+x)(1-3x+x^2)),{x,0,30}], x] (* or *) LinearRecurrence[{3,0,-3,1},{1,1,2,4},30] (* _Harvey P. Dale_, Dec 26 2013 *)
%o A166516 (Magma)
%o A166516 A166516:= func< n | (n mod 2)*Fibonacci(n)^2 +((n+1) mod 2)*Fibonacci(n-1)*Fibonacci(n+1) >;
%o A166516 [A166516(n): n in [0..40]]; // _G. C. Greubel_, Aug 03 2024
%o A166516 (SageMath)
%o A166516 f=fibonacci
%o A166516 def A166516(n): return (n%2)*f(n)^2 +((n+1)%2)*f(n-1)*f(n+1)
%o A166516 [A166516(n) for n in range(41)] # _G. C. Greubel_, Aug 03 2024
%Y A166516 Cf. A000045, A001654 (first differences), A166514.
%K A166516 easy,nonn
%O A166516 0,3
%A A166516 _Paul Barry_, Oct 16 2009