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 range(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.
KEYWORD
nonn,easy
AUTHOR
Rolf Pleisch, Feb 10 2008, Feb 14 2008
EXTENSIONS
More terms from Joshua Zucker, Feb 23 2008
STATUS
approved