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+1) = - A186679(2*n+1). - Reinhard Zumkeller, Feb 25 2011
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
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Apr 03 2003
EXTENSIONS
Simpler definition from Michael Somos, Mar 16 2004
STATUS
approved