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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

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