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A172012
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Expansion of (2-3*x)/(1-3*x-3*x^2) .
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0
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..25.
Wikipedia, Lucas sequence: Specific names.
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,3).
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FORMULA
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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]
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MATHEMATICA
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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 *)
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CROSSREFS
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Sequence in context: A203432 A151369 A143885 * A047014 A027519 A177012
Adjacent sequences: A172009 A172010 A172011 * A172013 A172014 A172015
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KEYWORD
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nonn,easy
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AUTHOR
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Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010
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EXTENSIONS
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Edited and extended by R. J. Mathar, Jan 23 2010
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STATUS
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approved
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