|
%I #17 Sep 23 2016 08:38:44
%S 0,1,2,9,24,81,234,729,2160,6561,19602,59049,176904,531441,1593594,
%T 4782969,14346720,43046721,129133602,387420489,1162241784,3486784401,
%U 10460294154,31381059609,94143001680,282429536481,847288078002,2541865828329,7625595890664
%N Expansion of x^2*(1-x)/((1-3*x)*(1-3*x^2)).
%C Limit(n->infinity) a(n+1)/a(n)=3.
%C 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.
%D Alain M. Robert, "Linear Algebra, Examples and Applications", World Scientific, 2005, p. 58.
%H Colin Barker, <a href="/A122006/b122006.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-9).
%F a(n) = 3*a(n-1) + 3*a(n-2) - 9*a(n-3). - _Philippe Deléham_, Mar 09 2009
%F From _Colin Barker_, Sep 23 2016: (Start)
%F a(n) = 3^(n-2) for n even.
%F a(n) = 3^(n-2)-3^((n-3)/2) for n odd. (End)
%t 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}]
%o (PARI) concat(0, Vec(x^2*(1-x)/((1-3*x)*(1-3*x^2)) + O(x^40))) \\ _Colin Barker_, Sep 23 2016
%Y Cf. A007179.
%K nonn,easy
%O 1,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 11 2006
|
