%I #10 Jul 01 2017 17:54:38

%S 0,1,2,8,25,77,236,721,2198,6698,20408,62173,189414,577055,1758012,

%T 5355828,16316664,49709120,151440064,461365896,1405562597,4282081163,

%U 13045466011,39743334364,121079049617,168870314746

%N Let X denote the 2 X 2 matrix [0,1; 1,exp(1)], let Y = X^n; a(n) = floor(Y[1,1]).

%C a(n)/a(n-1) tends to 3.046524695... = exp ArcSinh(e/2) = (exp(1)+(exp(2)+4)^(1/2))/2, the largest eigenvalue of X.

%e a(5) = 5 floor of term (1,1) in X^5 = 25.

%p ans:=[];

%p A:=Matrix(2,2,[[0,1],[1,exp(1)]]);

%p B:=Matrix(2,2,[[0,1],[1,exp(1)]]);

%p for n from 1 to 20 do

%p ans:=[op(ans),floor(B[1,1])];

%p B:=A.B;

%p od:

%p ans; # _N. J. A. Sloane_, Jul 01 2017

%K nonn

%O 1,3

%A _Gary W. Adamson_, Mar 30 2008

%E Definition corrected by _N. J. A. Sloane_, Jul 01 2017