OFFSET
0,2
COMMENTS
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
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*(4-x)/((1+x)*(x^2-3*x+1)). (End)
a(n) = A186679(2*n). - Reinhard Zumkeller, Feb 25 2011
a(-n) = - A128535(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
EXAMPLE
a(4) = 54 because F(4)*L(6) = 3*18.
G.f. = 4*x + 7*x^2 + 22*x^3 + 54*x^4 + 145*x^5 + 376*x^6 + 988*x^7 + ...
MAPLE
with(combinat); A128533:=n->fibonacci(2*n+2)+(-1)^(n+1); seq(A128533(k), k=0..50); # Wesley Ivan Hurt, Oct 19 2013
MATHEMATICA
LinearRecurrence[{2, 2, -1}, {0, 4, 7}, 40] (* Vincenzo Librandi, Feb 20 2013 *)
a[n_]:= Fibonacci[2n+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) vector(30, n, n--; fibonacci(2*(n+1)) + (-1)^(n+1)) \\ G. C. Greubel, Jan 07 2019
(Sage) [fibonacci(2*(n+1)) + (-1)^(n+1) for n in (0..30)] # G. C. Greubel, Jan 07 2019
(GAP) List([0..30], n -> Fibonacci(2*(n+1)) + (-1)^(n+1)); # G. C. Greubel, Jan 07 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Axel Harvey, Mar 08 2007
EXTENSIONS
More terms from Vincenzo Librandi, Feb 20 2013
STATUS
approved