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%I #16 Jun 26 2015 04:47:17
%S 2,3,15,54,207,783,2970,11259,42687,161838,613575,2326239,8819442,
%T 33437043,126769455,480619494,1822166847,6908359023,26191577610,
%U 99299809899,376474162527,1427321917278,5411388239415,20516130470079,77782556128482,294896059795683
%N Expansion of (2-3*x)/(1-3*x-3*x^2) .
%C The case k=3 in a family of sequences a(n) = L(k,n), L(k,n)=k*(L(k,n-1)+L(k,n-2)), L(k,0)=2 and L(k,1)=k.
%C The case k=1 is A000032 (classic Lucas sequence), k=2 is A080040, this here is essentially A085480.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,3).
%F a(n) = 3*( a(n-1)+a(n-2) ) = 2*A030195(n+1)-3*A030195(n).
%F L(k,n) = c^n+b^n where c=(k+d)/2 ; b=(k-d)/2; d=sqrt(k*(k+4)) (Binet formula).
%F a(0)=2, a(1)=3, a(n) = 3*a(n-1)+3*a(n-2). [_Harvey P. Dale_, Aug 24 2011]
%F a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 21*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%t CoefficientList[Series[(2-3x)/(1-3x-3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{3,3},{2,3},31] (* _Harvey P. Dale_, Aug 24 2011 *)
%K nonn,easy
%O 0,1
%A Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010
%E Edited and extended by _R. J. Mathar_, Jan 23 2010
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