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%I #51 Sep 08 2022 08:45:32
%S 0,1,6,38,240,1516,9576,60488,382080,2413456,15244896,96296288,
%T 608267520,3842197696,24269721216,153302722688,968355778560,
%U 6116740116736,38637152257536,244056393778688,1541612667187200
%N Generalized Fibonacci numbers: a(n) = 6*a(n-1) + 2*a(n-2).
%C 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
%H Joshua Zucker and Robert Israel, <a href="/A135030/b135030.txt">Table of n, a(n) for n = 0..1000</a> (n=0..51 from Joshua Zucker).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, 2).
%F a(0) = 0; a(1) = 1; a(n) = 2*(3*a(n-1) + a(n-2)).
%F a(n) = 1/(2*sqrt(11))*( (3 + sqrt(11))^n - (3 - sqrt(11))^n ).
%F G.f.: x/(1 - 6*x - 2*x^2). - _Harvey P. Dale_, Jun 20 2011
%F a(n+1) = Sum_{k=0..n} A099097(n,k)*2^k. - _Philippe Deléham_, Sep 16 2014
%F E.g.f.: (1/sqrt(11))*exp(3*x)*sinh(sqrt(11)*x). - _G. C. Greubel_, Sep 17 2016
%p A:= gfun:-rectoproc({a(0) = 0, a(1) = 1, a(n) = 2*(3*a(n-1) + a(n-2))},a(n),remember):
%p seq(A(n),n=1..30); # _Robert Israel_, Sep 16 2014
%t Join[{a=0,b=1},Table[c=6*b+2*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)
%t LinearRecurrence[{6,2},{0,1},30] (* or *) CoefficientList[Series[ -(x/(2x^2+6x-1)),{x,0,30}],x] (* _Harvey P. Dale_, Jun 20 2011 *)
%o (Sage) [lucas_number1(n,6,-2) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 24 2009
%o (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
%o (PARI) a(n)=([0,1; 2,6]^n*[0;1])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%Y 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.
%K nonn,easy
%O 0,3
%A _Rolf Pleisch_, Feb 10 2008, Feb 14 2008
%E More terms from _Joshua Zucker_, Feb 23 2008
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