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A135030 Generalized Fibonacci numbers: a(n) = 6*a(n-1) + 2*a(n-2). 9
0, 1, 6, 38, 240, 1516, 9576, 60488, 382080, 2413456, 15244896, 96296288, 608267520, 3842197696, 24269721216, 153302722688, 968355778560, 6116740116736, 38637152257536, 244056393778688, 1541612667187200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For n>0, a(n) equals the number of words of length n-1 over {0,1,...,7} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017

LINKS

Joshua Zucker and Robert Israel, Table of n, a(n) for n = 0..1000 (n=0..51 from Joshua Zucker).

Index entries for linear recurrences with constant coefficients, signature (6, 2).

FORMULA

a(0) = 0; a(1) = 1; a(n) = 2*(3*a(n-1) + a(n-2)).

a(n) = 1/(2*sqrt(11))*( (3 + sqrt(11))^n - (3 - sqrt(11))^n ).

G.f.: x/(1 - 6*x - 2*x^2). - Harvey P. Dale, Jun 20 2011

a(n+1) = Sum_{k=0..n} A099097(n,k)*2^k. - Philippe Deléham, Sep 16 2014

E.g.f.: (1/sqrt(11))*exp(3*x)*sinh(sqrt(11)*x). - G. C. Greubel, Sep 17 2016

MAPLE

A:= gfun:-rectoproc({a(0) = 0, a(1) = 1, a(n) = 2*(3*a(n-1) + a(n-2))}, a(n), remember):

seq(A(n), n=1..30); # Robert Israel, Sep 16 2014

MATHEMATICA

Join[{a=0, b=1}, Table[c=6*b+2*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)

LinearRecurrence[{6, 2}, {0, 1}, 30] (* or *) CoefficientList[Series[ -(x/(2x^2+6x-1)), {x, 0, 30}], x] (* Harvey P. Dale, Jun 20 2011 *)

PROG

(Sage) [lucas_number1(n, 6, -2) for n in xrange(0, 21)] # Zerinvary Lajos, Apr 24 2009

(MAGMA) [n le 2 select n-1 else 6*Self(n-1) + 2*Self(n-2): n in [1..35]]; // Vincenzo Librandi, Sep 18 2016

(PARI) a(n)=([0, 1; 2, 6]^n*[0; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

CROSSREFS

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A180250.

Sequence in context: A037499 A037676 A180030 * A217633 A162558 A215466

Adjacent sequences:  A135027 A135028 A135029 * A135031 A135032 A135033

KEYWORD

nonn,easy

AUTHOR

Rolf Pleisch, Feb 10 2008, Feb 14 2008

EXTENSIONS

More terms from Joshua Zucker, Feb 23 2008

STATUS

approved

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Last modified January 22 05:11 EST 2019. Contains 319353 sequences. (Running on oeis4.)