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a(n)=the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,3(n-1)].
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%I #2 Mar 30 2012 17:34:19

%S 1,3,38,783,22480,828000,37231704,1977187485,121098539008,

%T 8403438270285,651608685100000,55835951178466800,5239593453691293696,

%U 534383614812622168191,58857325474654519917440

%N a(n)=the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,3(n-1)].

%C The (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;1,1] is the Fibonacci number A000045(n).

%e a(4)=783 because if M is the 2 X 2 matrix [0,1;3,9], then M^4 is the 2 X 2 matrix [252,783,2349,7299].

%p with(linalg): a:=proc(n) local A,k: A[1]:=matrix(2,2,[0,1,n-1,3*(n-1)]): for k from 2 to n do A[k]:=multiply(A[k-1],A[1]) od: A[n][1,2] end: seq(a(n),n=1..18);

%t M[n_] = If[n > 1, MatrixPower[{{0, 1}, {n - 1, 3*(n - 1)}}, n], {{0, 1}, {1, 1}}] a = Table[M[n][[1, 2]], {n, 1, 50}]

%Y Cf. A000045, A000166.

%K nonn

%O 1,2

%A _Roger L. Bagula_, Jun 16 2005