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

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032, A000045, A003714, A212804.

Programs

GAP

List([0..40],n->2*Fibonacci(n-1)); # Muniru A Asiru, Oct 07 2018

Magma

[Lucas(n) - Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Sep 14 2014

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 *)

PARI

a(n)=fibonacci(n-1)<<1 \\ Charles R Greathouse IV, Jun 05 2011

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

Extensions

More terms from Philippe Deléham, Sep 20 2006

Corrected by T. D. Noe, Nov 01 2006