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%I #35 Jul 24 2021 02:31:49
%S 1,3,14,72,376,1968,10304,53952,282496,1479168,7745024,40553472,
%T 212340736,1111830528,5821620224,30482399232,159607914496,
%U 835717890048,4375875682304,22912382533632,119970792472576,628175224700928,3289168178315264,17222308171087872
%N Expansion of (1-3*x)/(1 - 6*x + 4*x^2).
%C Binomial transform of A001077. Second binomial transform of A084057. Third binomial transform of 1/(1-5*x^2). Let A=[1,1,1,1;3,1,-1,-3;3,-1,-1,3;1,-1,1,-1], the 4 X 4 Krawtchouk matrix. Then a(n)=trace((16(A*A`)^(-1))^n)/4.
%H Vincenzo Librandi, <a href="/A098648/b098648.txt">Table of n, a(n) for n = 0..300</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).
%F E.g.f.: exp(3*x)*cosh(sqrt(5)*x).
%F a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)/2.
%F a(n) = 2*(3*a(n-1) - 2*a(n-2)). - _Lekraj Beedassy_, Oct 22 2004
%F a(n) = A084326(n+1) - 3*A084326(n). - _R. J. Mathar_, Nov 10 2013
%F a(n) = 2^(n-1)*Lucas(2*n). - _Eric W. Weisstein_, Mar 31 2017
%t a[n_]:=(MatrixPower[{{5,1},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%t CoefficientList[Series[(1-3x)/(1-6x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,-4},{1,3},31] (* _Harvey P. Dale_, Jun 06 2011 *)
%t Table[2^(n - 1) LucasL[2 n], {n, 0, 20}] (* _Eric W. Weisstein_, Mar 31 2017 *)
%o (PARI) Vec((1-3*x)/(1 - 6*x + 4*x^2) + O(x^25)) \\ _Jinyuan Wang_, Jul 24 2021
%Y Cf. A098647.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Sep 18 2004
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