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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 October 19 15:50 EDT 2019. Contains 328223 sequences. (Running on oeis4.)