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A122006
Expansion of x^2*(1-x)/((1-3*x)*(1-3*x^2)).
3
0, 1, 2, 9, 24, 81, 234, 729, 2160, 6561, 19602, 59049, 176904, 531441, 1593594, 4782969, 14346720, 43046721, 129133602, 387420489, 1162241784, 3486784401, 10460294154, 31381059609, 94143001680, 282429536481, 847288078002, 2541865828329, 7625595890664
OFFSET
1,3
COMMENTS
Limit(n->infinity) a(n+1)/a(n)=3.
The sequence can be created by multiplying the n-th power of the matrix [[0,1,2],[1,2,0],[2,0,1]], multiplying from the right with the vector [1,0,0] and taking the middle element of the resulting vector.
REFERENCES
Alain M. Robert, "Linear Algebra, Examples and Applications", World Scientific, 2005, p. 58.
FORMULA
a(n) = 3*a(n-1) + 3*a(n-2) - 9*a(n-3). - Philippe Deléham, Mar 09 2009
From Colin Barker, Sep 23 2016: (Start)
a(n) = 3^(n-2) for n even.
a(n) = 3^(n-2)-3^((n-3)/2) for n odd. (End)
MATHEMATICA
M = {{0, 1, 2}, {1, 2, 0}, {2, 0, 1}} v[1] = {1, 0, 0} v[n_] := v[n] = M.v[n - 1] a1 = Table[v[n][[2]], {n, 1, 50}]
Rest[CoefficientList[Series[x^2(1-x)/((1-3x)(1-3x^2)), {x, 0, 30}], x]] (* or *) LinearRecurrence[{3, 3, -9}, {0, 1, 2}, 30] (* Harvey P. Dale, Aug 20 2024 *)
PROG
(PARI) concat(0, Vec(x^2*(1-x)/((1-3*x)*(1-3*x^2)) + O(x^40))) \\ Colin Barker, Sep 23 2016
CROSSREFS
Cf. A007179.
Sequence in context: A248436 A354016 A222667 * A200086 A143561 A363660
KEYWORD
nonn,easy
AUTHOR
STATUS
approved