OFFSET
0,3
COMMENTS
Generally, F(n)*L(n+k) = F(2*n + k) + F(k)*(-1)^(n+1):
For n>2, a(n) is twice the area of the triangle with vertices at (F(n-3), F(n-2)), (F(n-1), F(n)), and (L(n), L(n-1)), where F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, May 22 2014
a(n) is the maximum area of a quadrilateral with lengths of sides in order L(n-2), L(n-2), F(n), F(n) for n>2. - J. M. Bergot, Jan 28 2016
REFERENCES
Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
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
a(n) = F(2*(n-1)) - (-1)^(n+1), assuming F(0)=0 and L(0)=2.
From R. J. Mathar, Apr 16 2009: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: x*(-1+4*x)/((1+x)*(x^2-3*x+1)). (End)
a(n+1) = - A116697(2*n). - Reinhard Zumkeller, Feb 25 2011
a(-n) = - A128533(n). - Michael Somos, May 26 2014
0 = a(n)*(+4*a(n) + a(n+1) - 17*a(n+2)) + a(n+1)*(-14*a(n+1) + a(n+2)) + a(n+2)*(+4*a(n+2)) for all n in Z. - Michael Somos, May 26 2014
a(n) = ((-1)^n+(2^(-1-n)*(-(3-sqrt(5))^n*(3+sqrt(5))-(-3+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5)). - Colin Barker, Sep 28 2016
EXAMPLE
a(7) = 143 because F(7)*L(5) = 13*11.
G.f. = -x + 2*x^2 + 2*x^3 + 9*x^4 + 20*x^5 + 56*x^6 + 143*x^7 + ...
MAPLE
a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<0, -1, 2>>)[1, 1]:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 28 2016
MATHEMATICA
Table[Fibonacci[i]LucasL[i-2], {i, 0, 30}] (* Harvey P. Dale, Feb 16 2011 *)
LinearRecurrence[{2, 2, -1}, {0, -1, 2}, 40] (* Vincenzo Librandi, Feb 20 2013 *)
a[ n_] := Fibonacci[2 n - 2] + (-1)^n; (* Michael Somos, May 26 2014 *)
PROG
(Magma) [Fibonacci(n)*Lucas(n-2): n in [0..30]]; // Vincenzo Librandi, Feb 20 2013
(PARI) a(n)=([0, 1, 0; 0, 0, 1; -1, 2, 2]^n*[0; -1; 2])[1, 1] \\ Charles R Greathouse IV, Feb 01 2016
(PARI) a(n) = round(((-1)^n+(2^(-1-n)*(-(3-sqrt(5))^n*(3+sqrt(5))-(-3+sqrt(5))*(3+sqrt(5))^n))/sqrt(5))) \\ Colin Barker, Sep 28 2016
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Axel Harvey, Mar 09 2007
EXTENSIONS
More terms from Harvey P. Dale, Feb 16 2011
STATUS
approved