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 A172012 Expansion of (2-3*x)/(1-3*x-3*x^2) . 0
 2, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, 2326239, 8819442, 33437043, 126769455, 480619494, 1822166847, 6908359023, 26191577610, 99299809899, 376474162527, 1427321917278, 5411388239415, 20516130470079, 77782556128482, 294896059795683 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. The case k=1 is A000032 (classic Lucas sequence), k=2 is A080040, this here is essentially A085480. LINKS Wikipedia, Lucas sequence: Specific names. Index entries for linear recurrences with constant coefficients, signature (3,3). FORMULA a(n) = 3*( a(n-1)+a(n-2) ) = 2*A030195(n+1)-3*A030195(n). L(k,n) = c^n+b^n where c=(k+d)/2 ; b=(k-d)/2; d=sqrt(k*(k+4)) (Binet formula). a(0)=2, a(1)=3, a(n) = 3*a(n-1)+3*a(n-2). [Harvey P. Dale, Aug 24 2011] a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 21*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015 MATHEMATICA 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 *) CROSSREFS Sequence in context: A203432 A151369 A143885 * A047014 A027519 A245107 Adjacent sequences:  A172009 A172010 A172011 * A172013 A172014 A172015 KEYWORD nonn,easy AUTHOR Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010 EXTENSIONS Edited and extended by R. J. Mathar, Jan 23 2010 STATUS approved

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