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A118658 a(n) = 2*F(n-1) = L(n) - F(n), where F(n) and L(n) are Fibonacci and Lucas numbers respectively. 13
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 (list; graph; refs; listen; history; text; internal format)
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
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
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
Sequence in context: A273166 A331262 A300822 * A165912 A301823 A301999
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

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Last modified March 19 02:44 EDT 2024. Contains 370952 sequences. (Running on oeis4.)