A122006 Expansion of x^2*(1-x)/((1-3*x)*(1-3*x^2)).

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

Author

Roger L. Bagula and Gary W. Adamson, Sep 11 2006

Keywords

• nonn
• easy

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.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,3,-9).

Crossrefs

Cf. A007179.

Programs

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 *)

PARI

concat(0, Vec(x^2*(1-x)/((1-3*x)*(1-3*x^2)) + O(x^40))) \\ Colin Barker, Sep 23 2016

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)