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a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively.
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%I #81 Dec 31 2023 10:17:09

%S 2,0,2,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,

%T 8362,13530,21892,35422,57314,92736,150050,242786,392836,635622,

%U 1028458,1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634,78176338

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

%C Essentially the same as A006355, A047992, A054886, A055389, A068922, A078642, A090991. - _Philippe Deléham_, Sep 20 2006 and _Georg Fischer_, Oct 07 2018

%C Also the number of matchings in the (n-2)-pan graph. - _Eric W. Weisstein_, Jun 30 2016

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

%H Vincenzo Librandi, <a href="/A118658/b118658.txt">Table of n, a(n) for n = 0..1000</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ladder Graph.html">Ladder Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIndependentVertexSet.html">Maximal Independent Vertex Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalVertexCover.html">Minimal Vertex Cover</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PanGraph.html">Pan Graph</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).

%F From _Philippe Deléham_, Sep 20 2006: (Start)

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

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

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

%F a(n) = A006355(n) + 0^n. - _M. F. Hasler_, Nov 05 2014

%F a(n) = Lucas(n-2) + Fibonacci(n-2). - _Bruno Berselli_, May 27 2015

%F a(n) = 3*Fibonacci(n-2) + Fibonacci(n-5). - _Bruno Berselli_, Feb 20 2017

%F a(n) = 2*A212804(n). - _Bruno Berselli_, Feb 21 2017

%F 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

%p with(combinat): seq(2*fibonacci(n-1),n=0..40); # _Muniru A Asiru_, Oct 07 2018

%p a := n -> -2*I^n*ChebyshevU(n-2, -I/2):

%p seq(simplify(a(n)), n = 0..39); # _Peter Luschny_, Dec 03 2023

%t LinearRecurrence[{1, 1}, {2, 0}, 100] (* _Vladimir Joseph Stephan Orlovsky_, Jun 05 2011 *)

%t Table[LucasL[n] - Fibonacci[n], {n, 0, 40}] (* _Vincenzo Librandi_, Sep 14 2014 *)

%t Table[2 Fibonacci[n - 1], {n, 0, 20}] (* _Eric W. Weisstein_, Jun 30 2017 *)

%t 2 Fibonacci[Range[0, 20] - 1] (* _Eric W. Weisstein_, Jun 30 2017 *)

%t Subtract @@@ (Through[{LucasL, Fibonacci}[#]] & /@ Range[0, 20]) (* _Eric W. Weisstein_, Jun 30 2017 *)

%t CoefficientList[Series[(2 (-1 + x))/(-1 + x + x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jun 30 2017 *)

%o (PARI) a(n)=fibonacci(n-1)<<1 \\ _Charles R Greathouse IV_, Jun 05 2011

%o (Magma) [Lucas(n) - Fibonacci(n): n in [0..40]]; // _Vincenzo Librandi_, Sep 14 2014

%o (GAP) List([0..40],n->2*Fibonacci(n-1)); # _Muniru A Asiru_, Oct 07 2018

%Y Cf. A000032, A000045, A003714, A212804.

%K nonn,easy

%O 0,1

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

%E More terms from _Philippe Deléham_, Sep 20 2006

%E Corrected by _T. D. Noe_, Nov 01 2006