A128533 a(n) = F(n)*L(n+2) where F=Fibonacci and L=Lucas numbers.

0, 4, 7, 22, 54, 145, 376, 988, 2583, 6766, 17710, 46369, 121392, 317812, 832039, 2178310, 5702886, 14930353, 39088168, 102334156, 267914295, 701408734, 1836311902, 4807526977, 12586269024, 32951280100, 86267571271, 225851433718, 591286729878

Offset

0,2

Comments

Generally, F(n)*L(n+k) = F(2*n + k) + F(k)*(-1)^(n+1): if k = 0 then sequence is A001906, if k = 1 it is A081714.

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

Cf. A001906, A081714, A128534, A128535.

Sequence in context: A026548 A100098 A127361 * A162559 A126094 A073114

Adjacent sequences: A128530 A128531 A128532 * A128534 A128535 A128536

Keyword

nonn,easy

Author

Axel Harvey, Mar 08 2007

Extensions

More terms from Vincenzo Librandi, Feb 20 2013

Status

approved