%I A033304
%S A033304 2,6,11,26,57,129,289,650,1460,3281,7372,16565,37221,83635,187926,
%T A033304 422266,948823,2131986,4790529,10764221,24186985,54347662,122118088,
%U A033304 274396853,616564132,1385407029,3112981337,6994805571,15717185450
%N A033304 Expansion of (2+2*x-3*x^2)/(1-2*x-x^2+x^3).
%D A033304 R. P. Stanley, Enumerative Combinatorics I, p. 244, Eq. (36).
%D A033304 Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers 
               of the Seventh Order, Journal of Integer Sequences, Vol. 9 (2006), 
               Article 06.4.3.
%F A033304 a(n) = (1-2*cos(1/7*Pi))^(n+1)+(1+2*cos(2/7*Pi))^(n+1)+(1-2*cos(3/7*Pi))^(n+1). 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 27 2001
%F A033304 a(n) = trance of (n+1)-th power of the 3 X 3 matrix (in the example of 
               A066170): [1 1 1 / 1 1 0 / 1 0 0]. Alternatively, the sum of the 
               (n+1)-th powers of the roots of the corresponding characteristic 
               polynomial: -x^3 + 2x^2 + x -1 = 0. a(n) = A006356(n) + A006356(n-1) 
               + 2*A006356(n-2)/ where A006356 = 1, 3, 6, 14, 31, 70, 157... E.g. 
               a(3) = 26 = the trace of M^4. The characteristic polynomial of this 
               matrix (see A066170) is -x^3 + x^2 + x -1 and the roots are 2.24697960372..., 
               -.8019377358...and .55495813208...= a, b, c. Then Sum(a^4 + b^4 + 
               c^4) = 26. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 01 2004
%o A033304 (PARI) {a(n)=if(n<0, n=-n; polsym(x^3-x^2-2*x+1,n-1)[n], n+=2; polsym(1-x-2*x^2+x^3,
               n-1)[n])} /* Michael Somos Aug 03 2006 */
%Y A033304 Cf. A066170, A006356.
%Y A033304 A096975(n)=a(-1-n). - Michael Somos Aug 03 2006.
%Y A033304 Sequence in context: A079118 A034466 A007186 this_sequence A091622 A165821 
               A135048
%Y A033304 Adjacent sequences: A033301 A033302 A033303 this_sequence A033305 A033306 
               A033307
%K A033304 nonn
%O A033304 0,1
%A A033304 N. J. A. Sloane (njas(AT)research.att.com).