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A product of consecutive doubled Fibonacci numbers.
6

%I #23 Aug 03 2024 01:50:55

%S 1,1,2,4,10,25,65,169,442,1156,3026,7921,20737,54289,142130,372100,

%T 974170,2550409,6677057,17480761,45765226,119814916,313679522,

%U 821223649,2149991425,5628750625,14736260450,38580030724,101003831722

%N A product of consecutive doubled Fibonacci numbers.

%H G. C. Greubel, <a href="/A166516/b166516.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).

%F G.f.: (1-2*x-x^2+x^3) / ( (1-x)*(1+x)*(1-3*x+x^2) ).

%F a(n) = Fibonacci(2*floor(n/2) + 1)*Fibonacci(2*floor((n-1)/2) + 1).

%F a(n) = Fibonacci(A166514(2*n))^2 + Fibonacci(A166514(2*n+1))^2.

%F a(n) = Fibonacci(n)^2 * (1-(-1)^n)/2 + Fibonacci(n-1)*Fibonacci(n+1) * (1+(-1)^n)/2.

%F a(n+1)*a(n+3) - a(n+2)^2 = Fibonacci(n+2)^2 * (1-(-1)^n)/2.

%F a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - _G. C. Greubel_, May 15 2016

%t 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 (Magma)

%o A166516:= func< n | (n mod 2)*Fibonacci(n)^2 +((n+1) mod 2)*Fibonacci(n-1)*Fibonacci(n+1) >;

%o [A166516(n): n in [0..40]]; // _G. C. Greubel_, Aug 03 2024

%o (SageMath)

%o f=fibonacci

%o def A166516(n): return (n%2)*f(n)^2 +((n+1)%2)*f(n-1)*f(n+1)

%o [A166516(n) for n in range(41)] # _G. C. Greubel_, Aug 03 2024

%Y Cf. A000045, A001654 (first differences), A166514.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Oct 16 2009