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A167993
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Expansion of x^2/((3*x-1)*(3*x^2-1)).
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4
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0, 0, 1, 3, 12, 36, 117, 351, 1080, 3240, 9801, 29403, 88452, 265356, 796797, 2390391, 7173360, 21520080, 64566801, 193700403, 581120892, 1743362676, 5230147077, 15690441231, 47071500840, 141214502520, 423644039001, 1270932117003, 3812797945332, 11438393835996
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OFFSET
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0,4
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COMMENTS
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The terms satisfy a(n) = 3*a(n-1) +3*a(n-2) -9*a(n-3), so they follow the pattern a(n) = p*a(n-1) +q*a(n-2) -p*q*a(n-3) with p=q=3. This could be called the principal sequence for that recurrence because we have set all but one of the initial terms to zero. [p=q=1 leads to the principal sequence A004526. p=q=2 leads essentially to A032085. The common feature is that the denominator of the generating function does not have a root at x=1, so the sequences of higher order successive differences have the same recurrence as the original sequence. See A135094, A010036, A006516.]
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LINKS
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FORMULA
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a(2*n+1) = 3*a(2*n).
a(n) = 3*a(n-1) + 3*a(n-4) - 9*a(n-3).
G.f.: x^2/((3*x-1)*(3*x^2-1)).
a(n) = 3^(n-1)/2-3^(n/2-1)/2 for n even.
a(n) = 3^(n-1)/2-3^(n/2-1/2)/2 for n odd.
(End)
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MATHEMATICA
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CoefficientList[Series[x^2/((3*x - 1)*(3*x^2 - 1)), {x, 0, 50}], x] (* G. C. Greubel, Jul 03 2016 *)
LinearRecurrence[{3, 3, -9}, {0, 0, 1}, 30] (* Harvey P. Dale, Nov 05 2017 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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