%I
%S 0,2,2,2,5,8,13,25,45,81,150,275,504,928,1706,3136,5769,10610,19513,
%T 35891,66013,121415,223318,410745,755476,1389538,2555758,4700770,
%U 8646065,15902592,29249425,53798081,98950097,181997601,334745778
%N a(n) is the sum of the (1,2) and (1,3)entries of the matrix P^n + T^n, where the 3 X 3 matrices P and T are defined by P = [0,1,0; 0,0,1; 1,0,0] and T = [0,1,0; 0,0,1; 1,1,1].
%e a(7)=25 because
%e P^7 = [0,1,0; 0,0,1; 1,0,0],
%e T^7 = [7,11,13; 13,20,24; 24,37,44] and so
%e P^7 + T^7 = [7,12,13; 13,20,25; 25,37,44].
%p with(linalg): a:=proc(n) local P,T,v,k: P[1]:=matrix(3,3,[0,1,0,0,0,1,1,0,0]): T[1]:=matrix(3,3,[0,1,0,0,0,1,1,1,1]): v:=matrix(3,1,[0,1,1]): for k from 2 to n do P[k]:=multiply(P[1],P[k1]): T[k]:=multiply(T[1],T[k1]) od: evalm(P[n]+T[n])[1,2]+evalm(P[n]+T[n])[1,3] end: 0,seq(a(n),n=1..40);
%t v[0] = {0, 1, 1}; w[0] = {0, 1, 1}; M3 = {{0, 1, 0}, {0, 0, 1}, {1, 0, 0}}; Mt = {{0, 1, 0}, {0, 0, 1}, {1, 1, 1}}; v[n_] := v[n] = M3.v[n  1] w[n_] := w[n] = Mt.w[n  1] a = Table[(w[n] + v[n])[[1]], {n, 0, 50}]
%Y Cf. A000045, A000213.
%K nonn
%O 0,2
%A _Roger L. Bagula_, Jun 17 2005
