The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A081714 a(n) = F(n)*L(n+1) where F=Fibonacci and L=Lucas numbers. 6
 0, 3, 4, 14, 33, 90, 232, 611, 1596, 4182, 10945, 28658, 75024, 196419, 514228, 1346270, 3524577, 9227466, 24157816, 63245987, 165580140, 433494438, 1134903169, 2971215074, 7778742048, 20365011075, 53316291172, 139583862446, 365435296161, 956722026042 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also convolution of Fibonacci and Lucas numbers. 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 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,2,-1). FORMULA G.f.: x*(3-2*x)/((1+x)*(1-3*x+x^2)). a(n) = A122367(n) - (-1)^n. - R. J. Mathar, Jul 23 2010 a(n) = (L(n+1)^2 - F(2*n+2))/2 = ( A001254(n+1) - A001906(n+1) )/2. - Gary Detlefs, Nov 28 2010 a(n+1) = - A186679(2*n+1). - Reinhard Zumkeller, Feb 25 2011 a(n) = A035513(1,n-1)*A035513(2,n-1). - R. J. Mathar, Sep 04 2016 a(n)+a(n+1) = A005248(n+1). - R. J. Mathar, Sep 04 2016 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 MAPLE with(combinat): F:=n-> fibonacci(n): L:= n-> F(n+1)+F(n-1): a:= n-> F(n)*L(n+1): seq(a(n), n=0..30); MATHEMATICA Fibonacci[Range[0, 50]]*LucasL[Range[0, 50]+1] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011*) PROG (PARI) 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)", ")) (PARI) a(n)=fibonacci(n)*(fibonacci(n+2)+fibonacci(n)) (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 (MAGMA) [Fibonacci(n)*Lucas(n+1): n in [0..30]]; // Vincenzo Librandi, Sep 08 2012 (Sage) [fibonacci(n)*(fibonacci(n+2)+fibonacci(n)) for n in (0..30)] # G. C. Greubel, Jan 07 2019 (GAP) List([0..30], n -> Fibonacci(n)*(Fibonacci(n+2)+Fibonacci(n))); # G. C. Greubel, Jan 07 2019 CROSSREFS Cf. A000045, A000204. Sequence in context: A332270 A057433 A006074 * A117718 A268700 A349001 Adjacent sequences:  A081711 A081712 A081713 * A081715 A081716 A081717 KEYWORD nonn,easy AUTHOR Ralf Stephan, Apr 03 2003 EXTENSIONS Simpler definition from Michael Somos, Mar 16 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 2 11:44 EST 2021. Contains 349440 sequences. (Running on oeis4.)