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.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).
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
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 11 2006
STATUS
approved