A081714 - OEIS

%I #56 Jan 11 2025 19:56:05

%S 0,3,4,14,33,90,232,611,1596,4182,10945,28658,75024,196419,514228,

%T 1346270,3524577,9227466,24157816,63245987,165580140,433494438,

%U 1134903169,2971215074,7778742048,20365011075,53316291172,139583862446,365435296161,956722026042

%N a(n) = F(n)*L(n+1) where F=Fibonacci and L=Lucas numbers.

%C Also convolution of Fibonacci and Lucas numbers.

%C For n>2, a(n) represents twice the area of the triangle created by the three points (L(n-3), L(n-2)), (L(n-1), L(n)) and (F(n+3), F(n+2)) where L(k)=A000032(k) and F(k)=A000045(k). - _J. M. Bergot_, May 20 2014

%C For n>1, a(n) is the remainder when F(n+3)*F(n+4) is divided by F(n+1)*F(n+2). - _J. M. Bergot_, May 24 2014

%H Vincenzo Librandi, <a href="/A081714/b081714.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 (2,2,-1).

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

%F a(n) = A122367(n) - (-1)^n. - _R. J. Mathar_, Jul 23 2010

%F a(n) = (L(n+1)^2 - F(2*n+2))/2 = ( A001254(n+1) - A001906(n+1) )/2. - _Gary Detlefs_, Nov 28 2010

%F a(n+1) = - A186679(2*n+1). - _Reinhard Zumkeller_, Feb 25 2011

%F a(n) = A035513(1,n-1)*A035513(2,n-1). - _R. J. Mathar_, Sep 04 2016

%F a(n)+a(n+1) = A005248(n+1). - _R. J. Mathar_, Sep 04 2016

%F a(n) = (-(-1)^n+(2^(-1-n)*((3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5)). - _Colin Barker_, Sep 28 2016

%p with(combinat): F:=n-> fibonacci(n): L:= n-> F(n+1)+F(n-1):

%p a:= n-> F(n)*L(n+1): seq(a(n), n=0..30);

%t Fibonacci[Range[0,50]]*LucasL[Range[0,50]+1] (* _Vladimir Joseph Stephan Orlovsky_, Mar 17 2011*)

%o (PARI) my(x='x+O('x^51));for(n=0,50,print1(polcoeff(serconvol(Ser((1+2*x)/(1-x-x*x)),Ser(x/(1-x-x*x))),n)", "))

%o (PARI) a(n)=fibonacci(n)*(fibonacci(n+2)+fibonacci(n))

%o (PARI) a(n) = round((-(-1)^n+(2^(-1-n)*((3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/sqrt(5))) \\ _Colin Barker_, Sep 28 2016

%o (Magma) [Fibonacci(n)*Lucas(n+1): n in [0..30]]; // _Vincenzo Librandi_, Sep 08 2012

%o (Sage) [fibonacci(n)*(fibonacci(n+2)+fibonacci(n)) for n in (0..30)] # _G. C. Greubel_, Jan 07 2019

%o (GAP) List([0..30], n -> Fibonacci(n)*(Fibonacci(n+2)+Fibonacci(n))); # _G. C. Greubel_, Jan 07 2019

%Y Cf. A000045, A000204.

%K nonn,easy

%O 0,2

%A _Ralf Stephan_, Apr 03 2003

%E Simpler definition from _Michael Somos_, Mar 16 2004