%I #22 Jun 16 2016 23:27:48

%S 0,-3,-3,-9,-6,-24,-9,-66,-3,-189,57,-564,360,-1749,1644,-5607,6681,

%T -18465,25650,-62076,95415,-211878,348321,-731049,1256841,-2541468,

%U 4501572,-8881245,16046184,-31145307,57019797,-109482105,202204698,-385466112,716096199

%N a(n) = 3*a(n-2) - a(n-3), with a(0)=0, a(1)=a(2)=-3.

%C The Berndt-type sequence number 6 for the argument 2Pi/9 defined by the first relation from the section "Formula" below. Two sequences connected with a(n) (possessing the respective numbers 5 and 7) are discussed in A215664 and A215666 - for more details see comments to A215664 and Witula's reference. We have a(n) - a(n+1) = A215664(n).

%C From initial values and the recurrence formula we deduce that a(n)/3 are all integers.

%C We note that a(10) is the first element of a(n) which is positive integer and all (-1)^n*a(n+10) are positive integer, which can be obtained from the title recurrence relation.

%C The following decomposition holds (X - c(1)*c(2)^n)*(X - c(2)*c(4)^n)*(X - c(4)*c(1)^n) = X^3 - a(n)*X^2 - A215917(n-1)*X + (-1)^n.

%C If X(n) = 3*X(n-2) - X(n-3), n in Z, with X(n) = a(n) for every n=0,1,..., then X(-n) = abs(A215919(n)) = (-1)^n*A215919(n) for every n=0,1,...

%D R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math., (in press, 2012).

%D D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the nine order, (submitted, 2012).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0, 3, -1).

%F a(n) = = c(1)*c(2)^n + c(2)*c(4)^n + c(4)*c(1)^n, where c(j):=2*cos(2*Pi*j/9).

%F G.f.: -3*x*(1+x)/(1-3*x^2+x^3).

%e We have a(1)=a(2)=a(8)=-3, a(3)=a(6)=-9, a(4)+a(11)=-10*a(10), and 47*a(5)=2*a(11).

%t LinearRecurrence[{0,3,-1}, {0,-3,-3}, 50].

%o (PARI) concat(0,Vec(-3*(1+x)/(1-3*x^2+x^3)+O(x^99))) \\ _Charles R Greathouse IV_, Oct 01 2012

%Y Cf. A215455, A215634, A215635, A215636, A215664, A214699, A215007, A214683.

%K sign,easy

%O 0,2

%A _Roman Witula_, Aug 20 2012