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A128535 a(n) = F(n)*L(n-2) where F = Fibonacci and L = Lucas numbers.

%I A128535 #51 Sep 08 2022 08:45:30
%S A128535 0,-1,2,2,9,20,56,143,378,986,2585,6764,17712,46367,121394,317810,
%T A128535 832041,2178308,5702888,14930351,39088170,102334154,267914297,
%U A128535 701408732,1836311904,4807526975,12586269026,32951280098,86267571273,225851433716,591286729880
%N A128535 a(n) = F(n)*L(n-2) where F = Fibonacci and L = Lucas numbers.
%C A128535 Generally, F(n)*L(n+k) = F(2*n + k) + F(k)*(-1)^(n+1):
%C A128535 if k=0 the sequence is A001906, if k=1 it is A081714.
%C A128535 For n>2, a(n) is twice the area of the triangle with vertices at (F(n-3), F(n-2)), (F(n-1), F(n)), and (L(n), L(n-1)), where F(n)=A000045(n) and L(n)=A000032(n). - _J. M. Bergot_, May 22 2014
%C A128535 a(n) is the maximum area of a quadrilateral with lengths of sides in order L(n-2), L(n-2), F(n), F(n) for n>2. - _J. M. Bergot_, Jan 28 2016
%D A128535 Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
%H A128535 Vincenzo Librandi, <a href="/A128535/b128535.txt">Table of n, a(n) for n = 0..1000</a>
%H A128535 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A128535 a(n) = F(2*(n-1)) - (-1)^(n+1), assuming F(0)=0 and L(0)=2.
%F A128535 From _R. J. Mathar_, Apr 16 2009: (Start)
%F A128535 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A128535 G.f.: x*(-1+4*x)/((1+x)*(x^2-3*x+1)). (End)
%F A128535 a(n+1) = - A116697(2*n). - _Reinhard Zumkeller_, Feb 25 2011
%F A128535 a(-n) = - A128533(n). - _Michael Somos_, May 26 2014
%F A128535 0 = a(n)*(+4*a(n) + a(n+1) - 17*a(n+2)) + a(n+1)*(-14*a(n+1) + a(n+2)) + a(n+2)*(+4*a(n+2)) for all n in Z. - _Michael Somos_, May 26 2014
%F A128535 a(n) = ((-1)^n+(2^(-1-n)*(-(3-sqrt(5))^n*(3+sqrt(5))-(-3+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5)). - _Colin Barker_, Sep 28 2016
%e A128535 a(7) = 143 because F(7)*L(5) = 13*11.
%e A128535 G.f. = -x + 2*x^2 + 2*x^3 + 9*x^4 + 20*x^5 + 56*x^6 + 143*x^7 + ...
%p A128535 a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<0,-1,2>>)[1, 1]:
%p A128535 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 28 2016
%t A128535 Table[Fibonacci[i]LucasL[i-2], {i,0,30}] (* _Harvey P. Dale_, Feb 16 2011 *)
%t A128535 LinearRecurrence[{2, 2, -1}, {0, -1, 2}, 40] (* _Vincenzo Librandi_, Feb 20 2013 *)
%t A128535 a[ n_] := Fibonacci[2 n - 2] + (-1)^n; (* _Michael Somos_, May 26 2014 *)
%o A128535 (Magma) [Fibonacci(n)*Lucas(n-2): n in [0..30]]; // _Vincenzo Librandi_, Feb 20 2013
%o A128535 (PARI) a(n)=([0,1,0; 0,0,1; -1,2,2]^n*[0;-1;2])[1,1] \\ _Charles R Greathouse IV_, Feb 01 2016
%o A128535 (PARI) a(n) = round(((-1)^n+(2^(-1-n)*(-(3-sqrt(5))^n*(3+sqrt(5))-(-3+sqrt(5))*(3+sqrt(5))^n))/sqrt(5))) \\ _Colin Barker_, Sep 28 2016
%Y A128535 Cf. A001906, A081714, A128533, A128534.
%K A128535 sign,easy
%O A128535 0,3
%A A128535 _Axel Harvey_, Mar 09 2007
%E A128535 More terms from _Harvey P. Dale_, Feb 16 2011