OFFSET
0,4
COMMENTS
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.]
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).
FORMULA
a(2*n+1) = 3*a(2*n).
a(2*n) = A122006(2*n)/2.
a(n) = 3*a(n-1) + 3*a(n-4) - 9*a(n-3).
a(n+1) - a(n) = A122006(n).
a(n) = (3^n - A108411(n+1))/6.
G.f.: x^2/((3*x-1)*(3*x^2-1)).
From Colin Barker, Sep 23 2016: (Start)
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)
MATHEMATICA
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 *)
PROG
(PARI) Vec(x^2/((3*x-1)*(3*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 29 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Nov 16 2009
EXTENSIONS
Formulae corrected by Johannes W. Meijer, Jun 28 2011
STATUS
approved