%I #38 Sep 08 2022 08:44:40
%S 0,1,6,47,348,2605,19458,145403,1086456,8118169,60660030,453260039,
%T 3386820564,25306783813,189095729082,1412948996435,10557746998512,
%U 78888920951857,589468742694774,4404590586639071,32911699689476940
%N Expansion of x/(1 - 6*x - 11*x^2).
%C Let the generator matrix for the binary Golay G_24 code be [I|B]. Then a(n)=(A^n)_1,2 for instance. Third binomial transform of (0,1,0,20,0,400,0,8000,...). - _Paul Barry_, Feb 13 2004
%H Vincenzo Librandi, <a href="/A015553/b015553.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,11).
%F a(n) = 6*a(n-1) + 11*a(n-2).
%F a(n) = (1/4)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*4^k. - _Benoit Cloitre_, Oct 25 2003
%F a(n) = sqrt(5)(3 + 2*sqrt(5))^n/20 - sqrt(5)(3 - 2*sqrt(5))^n/20. - _Paul Barry_, Feb 13 2004
%F E.g.f.: (exp(x*(3 + 2*sqrt(5))) - exp(x*(3 - 2*sqrt(5))))/(4*sqrt(5)). - _Iain Fox_, Dec 31 2017
%t a[n_]:=(MatrixPower[{{1,4},{1,-7}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)
%t LinearRecurrence[{6, 11}, {0, 1}, 30] (* _Vincenzo Librandi_, Nov 13 2012 *)
%o (Sage) [lucas_number1(n,6,-11) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 24 2009
%o (Magma) [n le 2 select n-1 else 6*Self(n-1) + 11*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 13 2012
%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1 - 6*x - 11*x^2))) \\ _G. C. Greubel_, Dec 30 2017
%Y Cf. A015551.
%K nonn,easy
%O 0,3
%A _Olivier Gérard_