A118658 a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.

2, 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset

0,1

Comments

Essentially the same as A006355, A047992, A054886, A055389, A068922, A078642, A090991. - Philippe Deléham, Sep 20 2006 and Georg Fischer, Oct 07 2018

Also the number of matchings in the (n-2)-pan graph. - Eric W. Weisstein, Jun 30 2016

Also the number of maximal independent vertex sets (and minimal vertex covers) in the (n-1)-ladder graph. - Eric W. Weisstein, Jun 30 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Independent Edge Set

Eric Weisstein's World of Mathematics, Ladder Graph

Eric Weisstein's World of Mathematics, Matching

Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Eric Weisstein's World of Mathematics, Minimal Vertex Cover

Eric Weisstein's World of Mathematics, Pan Graph

Index entries for linear recurrences with constant coefficients, signature (1,1).

Formula

From Philippe Deléham, Sep 20 2006: (Start)

a(0)=2, a(1)=0; for n > 1, a(n) = a(n-1) + a(n-2).

G.f. (2 - 2*x)/(1 - x - x^2).

a(0)=2 and a(n) = 2*A000045(n-1) for n > 0. (End)

a(n) = A006355(n) + 0^n. - M. F. Hasler, Nov 05 2014

a(n) = Lucas(n-2) + Fibonacci(n-2). - Bruno Berselli, May 27 2015

a(n) = 3*Fibonacci(n-2) + Fibonacci(n-5). - Bruno Berselli, Feb 20 2017

a(n) = 2*A212804(n). - Bruno Berselli, Feb 21 2017

E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Apr 18 2022

Maple

with(combinat): seq(2*fibonacci(n-1), n=0..40); # Muniru A Asiru, Oct 07 2018
a := n -> -2*I^n*ChebyshevU(n-2, -I/2):
seq(simplify(a(n)), n = 0..39);  # Peter Luschny, Dec 03 2023

Mathematica

LinearRecurrence[{1, 1}, {2, 0}, 100] (* Vladimir Joseph Stephan Orlovsky, Jun 05 2011 *)
Table[LucasL[n] - Fibonacci[n], {n, 0, 40}] (* Vincenzo Librandi, Sep 14 2014 *)
Table[2 Fibonacci[n - 1], {n, 0, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
2 Fibonacci[Range[0, 20] - 1] (* Eric W. Weisstein, Jun 30 2017 *)
Subtract @@@ (Through[{LucasL, Fibonacci}[#]] & /@ Range[0, 20]) (* Eric W. Weisstein, Jun 30 2017 *)
CoefficientList[Series[(2 (-1 + x))/(-1 + x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 30 2017 *)

Prog

(PARI) a(n)=fibonacci(n-1)<<1 \\ Charles R Greathouse IV, Jun 05 2011
(Magma) [Lucas(n) - Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Sep 14 2014
(GAP) List([0..40], n->2*Fibonacci(n-1)); # Muniru A Asiru, Oct 07 2018

Crossrefs

Cf. A000032, A000045, A003714, A212804.

Sequence in context: A273166 A331262 A300822 * A165912 A301823 A301999

Adjacent sequences: A118655 A118656 A118657 * A118659 A118660 A118661

Keyword

nonn,easy

Author

Bill Jones (b92057(AT)yahoo.com), May 18 2006

Extensions

More terms from Philippe Deléham, Sep 20 2006

Corrected by T. D. Noe, Nov 01 2006

Status

approved