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A099015 a(n) = Fib(n+1)*(2*Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2).

%I A099015 #23 Jul 04 2025 18:41:50
%S A099015 1,2,8,33,140,592,2509,10626,45016,190685,807764,3421728,14494697,
%T A099015 61400482,260096680,1101787113,4667245276,19770767984,83750317589,
%U A099015 354772037730,1502838469496,6366125914117,26967342128548
%N A099015 a(n) = Fib(n+1)*(2*Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2).
%C A099015 Form the matrix A=[1,1,1,1;3,2,1,0;3,1,0,0;1,0,0,0]=(binomial(3-j,i)). Then a(n)=(2,2)-element of A^n.
%H A099015 G. C. Greubel, <a href="/A099015/b099015.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..172 from Vincenzo Librandi)
%H A099015 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-3,-1).
%F A099015 G.f.: (1-x-4*x^2)/((1+x-x^2)*(1-4*x-x^2)).
%F A099015 G.f.: (1-x-4*x^2)/(1-3*x-6*x^2+3*x^3+x^4).
%F A099015 a(n) = (3*Fib(3*n+1) + (-1)^n*Fib(n-3))/5.
%F A099015 a(n) = (2+sqrt(5))^n*(3/10 + 3*sqrt(5)/50) + (2-sqrt(5))^n*(3/10 - 3*sqrt(5)/50) + (-1)^n*((1/2 - sqrt(5)/2)^n*(1/5 + 2*sqrt(5)/25) + (1/5 - 2*sqrt(5)/25)*(1/2 + sqrt(5)/2)^n).
%t A099015 LinearRecurrence[{3,6,-3,-1},{1,2,8,33},30] (* _Harvey P. Dale_, Nov 28 2015 *)
%t A099015 CoefficientList[Series[(1-x-4*x^2)/((1+x-x^2)*(1-4*x-x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 31 2017 *)
%o A099015 (Magma) [Fibonacci(n+1)*(2*Fibonacci(n)^2 + Fibonacci(n)*Fibonacci(n-1) + Fibonacci(n-1)^2): n in [0..30]]; // _Vincenzo Librandi_, Jun 05 2011
%o A099015 (PARI) a(n)=my(e=fibonacci(n-1),f=fibonacci(n));(e+f)*(2*f^2+f*e+e^2) \\ _Charles R Greathouse IV_, Jun 05 2011
%o A099015 (PARI) first(n) = Vec((1 - x - 4*x^2)/(1 - 3*x - 6*x^2 + 3*x^3 + x^4) + O(x^n)) \\ _Iain Fox_, Dec 31 2017
%Y A099015 Cf. A000045, A056570, A066258, A066259, A099014, A033887.
%K A099015 easy,nonn
%O A099015 0,2
%A A099015 _Paul Barry_, Sep 22 2004