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a(n)=the sum of the (1,2)- and (1,3)-entries and twice the (1,4)-entry of the matrix P^n + T^n, where the 4 X 4 matrices P and T are defined by P=[0,1,0,0;0,0,1,0;0,0,0,1;1,0,0,0] and T=[0,1,0,0;0,0,1,0;0,0,0,1;1,0,0,1].
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%I #7 Jun 13 2015 00:51:52

%S 0,2,2,4,2,4,5,8,8,12,16,23,29,41,56,78,105,146,201,278,381,527,727,

%T 1004,1383,1910,2636,3639,5020,6930,9565,13203,18221,25151,34715,

%U 47917,66136,91287,126001,173917,240051,331338,457338,631254,871303,1202641

%N a(n)=the sum of the (1,2)- and (1,3)-entries and twice the (1,4)-entry of the matrix P^n + T^n, where the 4 X 4 matrices P and T are defined by P=[0,1,0,0;0,0,1,0;0,0,0,1;1,0,0,0] and T=[0,1,0,0;0,0,1,0;0,0,0,1;1,0,0,1].

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-1,0,0,-1).

%F G.f.: (-3*x^6 - x^5 - 2*x^4 - 2*x^3 + 2*x^2 + 2)/(x^8 + x^5 - 2*x^4 - x + 1).

%e a(7)=8 because P^7=[0,0,0,1;1,0,0,0;0,1,0,0;0,0,1,0], T^7=[1,1,1,2;2,1,1,3;3,2,1,4;4,3,2,5] and so P^7+T^7=[1,1,1,3;3,1,1,3;3,3,1,4;4,3,3,5] and now a(7)=1+1+2*3=8.

%p with(linalg): a:=proc(n) local P,T,k: P[1]:=matrix(4,4,[0,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0]): T[1]:=matrix(4,4,[0,1,0,0,0,0,1,0,0,0,0,1,1,0,0,1]):for k from 2 to n do P[k]:=multiply(P[1],P[k-1]): T[k]:=multiply(T[1],T[k-1]) od: evalm(P[n]+T[n])[1,2]+evalm(P[n]+T[n])[1,3]+2*evalm(P[n]+T[n])[1,4] end: 0,seq(a(n),n=1..50);

%t v[0] = {0, 1, 1, 2}; w[0] = {0, 1, 1, 2}; M4 = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 0}}; Mt = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 1}}; v[n_] := v[n] = M4.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, A000931.

%K nonn,easy

%O 0,2

%A _Roger L. Bagula_, Jun 17 2005