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A143212 a(n) = Fibonacci(n) * (Fibonacci(n+2) - 1).

%I A143212 #21 Jul 22 2024 15:17:49
%S A143212 1,2,8,21,60,160,429,1134,2992,7865,20648,54144,141897,371722,973560,
%T A143212 2549421,6675460,17478176,45761045,119808150,313668576,821205937,
%U A143212 2149962768,5628704256,14736185425,38579909330,101003635304
%N A143212 a(n) = Fibonacci(n) * (Fibonacci(n+2) - 1).
%C A143212 Lim_{n -> oo} a(n)/a(n-1) tends to phi^2.
%C A143212 a(n) = Product of sum of first n Fibonacci numbers and Fibonacci number(n). - _Vladimir Joseph Stephan Orlovsky_, Oct 13 2009
%H A143212 G. C. Greubel, <a href="/A143212/b143212.txt">Table of n, a(n) for n = 1..1000</a>
%H A143212 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-5,-1,1).
%F A143212 a(n) = A000045(n) * A000071(n+2).
%F A143212 a(n) = Sum_{k=1..n} A143211(n, k) (row sums of A143211).
%F A143212 From _R. J. Mathar_, Sep 06 2008: (Start)
%F A143212 G.f.: (1-x+x^2)/((1+x)*(1-3*x+x^2)*(1-x-x^2)).
%F A143212 a(n) = (-5*A000045(n+1) + 3*(-1)^n + 7*A001906(n+1) -3*A001906(n))/5. (End)
%F A143212 a(n) = Fibonacci(n)*Sum_{k=0..n} Fibonacci(k). - _Paul Barry_, Jan 05 2009
%e A143212 a(5) = 60 = F(5) * (F(7)-1) = 5*12.
%e A143212 a(5) = 60 = sum of row 5 terms of triangle A143211: (5 + 5 + 10 + 15 + 25).
%t A143212 LinearRecurrence[{3,1,-5,-1,1}, {1,2,8,21,60}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Oct 13 2009 *)
%t A143212 Table[Fibonacci[n](Fibonacci[n+2]-1),{n,30}] (* _Harvey P. Dale_, Dec 14 2012 *)
%o A143212 (Magma) [Fibonacci(n)*(Fibonacci(n+2)-1): n in [1..40]]; // _G. C. Greubel_, Jul 21 2024
%o A143212 (SageMath) [fibonacci(n)*(fibonacci(n+2)-1) for n in range(1,41)] # _G. C. Greubel_, Jul 21 2024
%Y A143212 Cf. A000045, A000071, A143211.
%K A143212 nonn
%O A143212 1,2
%A A143212 _Gary W. Adamson_, Jul 30 2008