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%I #14 Sep 08 2022 08:45:52
%S 1,3,12,56,280,1440,7488,39104,204544,1070592,5604864,29345792,
%T 153653248,804532224,4212572160,22057287680,115493404672,604731211776,
%U 3166413520896,16579556016128,86811681488896,454551863820288
%N Hankel transform of A176280.
%H G. C. Greubel, <a href="/A176281/b176281.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16,8).
%F G.f.: (1-5*x+4*x^2)/(1-8*x+16*x^2-8*x^3) = (1-5*x+4*x^2)/((1-2*x)*(1-6*x+4*x^2)).
%F a(n) = 2^(n-1) + (3-sqrt(5))^n*((5-sqrt(5))/20) + (3+sqrt(5))^n*((5+sqrt(5))/20).
%F a(n) = 2^(n-1) + A082761(n)/2. - _R. J. Mathar_, Sep 30 2012
%F a(0)=1, a(1)=3, a(2)=12, a(n) = 8*a(n-1) - 16*a(n-2) + 8*a(n-3). - _Harvey P. Dale_, Aug 14 2013
%F a(n) = 2^(n-1)*(Fibonacci(2*n+1) + 1). - _G. C. Greubel_, Nov 24 2019
%p with(combinat); seq(2^(n-1)*(fibonacci(2*n+1) + 1), n=0..30); # _G. C. Greubel_, Nov 24 2019
%t CoefficientList[Series[(1-5x+4x^2)/((1-2x)(1-6x+4x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{8,-16,8},{1,3,12},40] (* _Harvey P. Dale_, Aug 14 2013 *)
%o (PARI) vector(31, n, 2^(n-2)*(fibonacci(2*n-1) + 1)) \\ _G. C. Greubel_, Nov 24 2019
%o (Magma) [2^(n-1)*(Fibonacci(2*n+1) + 1): n in [0..30]]; _G. C. Greubel_, Nov 24 2019
%o (Sage) [2^(n-1)*(fibonacci(2*n+1) + 1) for n in (0..30)] # _G. C. Greubel_, Nov 24 2019
%o (GAP) List([0..30], n-> 2^(n-1)*(Fibonacci(2*n+1) + 1)); # _G. C. Greubel_, Nov 24 2019
%Y Cf. A000045.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 14 2010
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