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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. 11
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, A090991. - Philippe Deléham, Sep 20 2006

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 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, 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) = F(n) + F(n+3) with n>=-3, where F = A000045. - Zerinvary Lajos, Jan 31 2008

Closed form: a(n) = ((1/2)+(1/2)*sqrt(5))^n-(1/5)*((1/2)+(1/2)*sqrt(5))^n*sqrt(5)+(1/5)*sqrt(5)*((1/2)-(1/2) *sqrt(5))^n+((1/2)-(1/2)*sqrt(5))^n. - Paolo P. Lava, Nov 19 2008

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

MAPLE

BB := n->if n=0 then 2; > elif n=1 then 0; > else BB(n-2)+BB(n-1); > fi: > L:=[]: for k from 0 to 38 do L:=[op(L), BB(k)]: od: L; - Zerinvary Lajos, Mar 19 2007

with(combinat): seq(fibonacci(n)+fibonacci(n+3), n=-3..35); - Zerinvary Lajos, Jan 31 2008

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

CROSSREFS

Cf. A000032, A000045, A003714, A212804.

Sequence in context: A002121 A279158 A273166 * A165912 A171936 A071055

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

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Last modified July 23 05:58 EDT 2017. Contains 289686 sequences.