A081714 a(n) = F(n)*L(n+1) where F=Fibonacci and L=Lucas numbers.

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

Offset: 0

Author

Ralf Stephan, Apr 03 2003

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000045, A000204.

Programs

GAP

List([0..30], n -> Fibonacci(n)*(Fibonacci(n+2)+Fibonacci(n))); # G. C. Greubel, Jan 07 2019

Magma

[Fibonacci(n)*Lucas(n+1): n in [0..30]]; // Vincenzo Librandi, Sep 08 2012

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*)

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)", "))

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

Sage

[fibonacci(n)*(fibonacci(n+2)+fibonacci(n)) for n in (0..30)] # G. C. Greubel, Jan 07 2019

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

Extensions

Simpler definition from Michael Somos, Mar 16 2004