A128535 a(n) = F(n)*L(n-2) where F = Fibonacci and L = Lucas numbers.

0, -1, 2, 2, 9, 20, 56, 143, 378, 986, 2585, 6764, 17712, 46367, 121394, 317810, 832041, 2178308, 5702888, 14930351, 39088170, 102334154, 267914297, 701408732, 1836311904, 4807526975, 12586269026, 32951280098, 86267571273, 225851433716, 591286729880

Offset

0,3

Comments

Generally, F(n)*L(n+k) = F(2*n + k) + F(k)*(-1)^(n+1):

if k=0 the sequence is A001906, if k=1 it is A081714.

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

Cf. A001906, A081714, A128533, A128534.

Sequence in context: A007024 A019223 A192302 * A180753 A220971 A368182

Adjacent sequences: A128532 A128533 A128534 * A128536 A128537 A128538

Keyword

sign,easy

Author

Axel Harvey, Mar 09 2007

Extensions

More terms from Harvey P. Dale, Feb 16 2011

Status

approved